When Dominic Ryan and Isaac Boss pounced for first half tries, it looked like Leinster, whose scrum had been proving dominant, would go on to claim their first win at Firhill Stadium in over four years.
But the province's stranglehold on the match was loosened in the third quarter as gutsy Glasgow began to make in-roads.
Missing eight of their frontline players and trailing 16-6 at half-time, the Warriors closed the gap to seven points while Leinster replacement John Fogarty was in the sin-bin.
They were back on terms when Canadian international DTH van der Merwe got a try his industry deserved. The 19-year-old Weir, only minutes on the pitch, added the levelling conversion with great aplomb.
As errors and indiscipline left Leinster struggling for possession, Glasgow engineered a drop goal chance for Weir and he took it, moving the hosts 19-16 ahead.
Number 10 Isa Nacewa looked to have secured a share of the spoils for Leinster when he smacked over a terrific penalty from the left.
But with time running out, a costly penalty at a lineout saw Weir boot the Warriors clear again. Nacewa drew a last-minute drop goal attempt wide of the posts, leaving Sean Lineen's side to celebrate at the final whistle.
A couple of key elements from last season's Glasgow team were missing for this league opener, with top points scorer Dan Parks having made the move to the Cardiff Blues and the 'Killer Bs' back row broken up by Kelly Brown's switch to Saracens and injuries to John Barclay and Johnnie Beattie.
Ruaridh Jackson was joined at half-back by tournament newcomer Henry Pyrgos, and Ryan Wilson, Calum Forrester and Richie Vernon combined in a new-look back row. Argentinian full-back Bernardo Stortoni took on the captaincy role, in the absence of Alastair Kellock and Chris Cusiter.
For New Zealander Joe Schmidt's first competitive game in charge, Leinster were captained by young flanker Sean O'Brien who is back fully fit after breaking his leg in March. Full-back Luke Fitzgerald also returned from a long lay-off, his last competitive game being Ireland's Test against Australia last November.
New signings Isaac Boss, Heinke van der Merwe and Ed O'Donoghue made their senior bows for the province, while prop Simon Shawe took the step-up from All-Ireland League level.
Nacewa and Fitzgerald had early runs before Glasgow were penalised for coming in at the side of a ruck, allowing centre Fergus McFadden to kick Leinster into a third minute lead.
A Thom Evans burst got the Warriors within scoring range off the restart and Jackson punished a Leinster offside with his first points of the match.
The action was error strewn at times and when a Glasgow attacking move broke down, Leinster sensed their first opening. Centre Eoin O'Malley hacked on and was unfortunate to be penalised for a knock on before Shane Horgan edged towards the try-line.
Glasgow recovered from a shaky scrum in their 22 and a tremendous touchfinder from Jackson gave them the territory they needed.
Still, Leinster looked the more creative and when Jamie Heaslip broke from a scrum and offloaded to the supporting Boss, the visitors covered 60 metres in the space of a few seconds. Glasgow conceded a penalty which McFadden converted for 6-3.
Four minutes later, Glasgow skipper Stortoni had to be sharp to clear up as Nacewa went the aerial route, hanging the ball up for young winger Niall Morris to chase. The Warriors lost possession, and from the ensuing scrum, Dominic Ryan had a try ruled out for a forward pass from Heaslip.
A superb penalty from Jackson, from wide out on the right, showed his place-kicking ability, but the remainder of the first half largely went Leinster's way.
Glasgow's set pieces difficulties transferred to the lineout in the 26th minute, and the visitors took full advantage.
Fergus Thomson threw long at a short lineout, Heaslip read it and bounded forward into the Glasgow half before passing to the left for Ryan to gather and impressively beat the cover to the try-line.
McFadden's conversion attempt was off target, and Glasgow, with Pyrgos passing well, managed to build through the phases. A neat incision by van der Merwe, coming off his wing, forced Leinster to scramble back towards their whitewash. However, a subsequent knock on spoilt a great position for Lineen's men.
From a Glasgow scrum, Leinster manufactured their second try. The Scots were again disrupted in the set piece, O'Brien kicked the ball free and Boss chipped forward for Heaslip to chase. Glasgow's rearguard wilted under the pressure, with Evans and Federico Aramburu failing to clear the danger and the advancing Boss stretched out his arm for the touchdown.
The try went unconverted and Glasgow enjoyed a decent spell in the minutes leading up to half-time, helped by strong runs from Graeme Morrison and Richie Gray. A try went a-beginning, nevertheless, when a well-worked short lineout from hooker Thomson was called back for a crooked throw.
It was a gradual turnaround in the second half. It started with a purposeful maul from the Leinster forwards, but the third quarter was bossed by Glasgow - the hard work put in by Calum Forrester, Tom Ryder and Vernon forcing the visitors onto the back foot.
van der Merwe continued to look lively out wide, and there were cracks beginning to appear in Leinster's armoury. Replacement hooker John Fogarty was barely a minute on the pitch when he got involved in a scuffle with Stortoni and was yellow carded for punching.
Jackson missed the resulting penalty. He converted an easier opportunity, from almost in front of the posts in the 55th minute, to close the gap - a powerful surge from number 8 Vernon was the key play in the build-up.
Leinster were losing their discipline at the breakdown, and although Jackson missed a shot from the ten-metre line, the momentum was clearly with the Warriors.
As Lineen dug into his bench, the vocal home crowd really got behind Glasgow and some slick passing paved the way for Weir to send big winger van der Merwe crashing over in the right corner, past the despairing Boss.
Weir, who came on for Jackson, followed up with a fine conversion, and carries from Richie Gray and Vernon helped create the space for the Scotland Under-20 international to clip a drop goal over, suddenly making it 19-16.
Nacewa's penalty in return was a high quality score and for a short while, it seemed that the game was destined to end in a draw.
A lineout infringement offered Weir the chance to kick his side ahead again and he coolly did so, making it three years in-a-row that Glasgow have won their opening league fixture.
Summing up his feelings on the game, new Leinster supremo Schmidt said: "It was certainly a game that got away from us after half-time.
"We're really disappointed to snatch defeat from the jaws of victory and it was a case of us making errors which enabled them to keep the pressure on (in the second half).
"There were moments in the game when we had chances; a forward pass that was given which denied us a try.
"I was really pleased for Dom (Ryan) and I thought he played very well. I didn't know him before I arrived but he has performed well during pre-season and carried that into tonight.
"The boys are very frustrated in there and they feel as though they let themselves down. It's up to us now to take it on the chin and get ourselves in order for next week."
