Ray Ofisa and Brian Tuohy both scored converted tries in the run-up to half-time to fire the underdogs into a deserved 20-8 lead after Devin Toner had crossed first for the hosts.
With Johnny O'Connor hurtling into tackles and Niall O'Connor a steadying presence at out-half, Connacht were on course for their first away win over Leinster since September 2002.
But just as they did against Aironi seven days ago, Leinster managed to turn things around after the break.
A succession of penalty goals - Isa Nacewa finished with 14 points - and Luke Fitzgerald's 65th-minute try edged the hosts ahead, and a last-minute drop goal from Ian Madigan even took the losing bonus point away from Connacht.
Eric Elwood's men were first to threaten, Niall O'Connor missing a penalty from inside his own half with the wind behind him.
Clever work from Mike McCarthy and Johnny O'Connor led to Niall O'Connor kicking to the right for Mark McCrea to gather with the try-line in his sight, but he was soon crowded out.
Connacht maintained their presence in the hosts' 22 and O'Connor duly drilled a ninth-minute penalty through the uprights. That score only served to spark Leinster into life.
Their stand-in captain Rhys Ruddock barged through the middle and over halfway and, with Connacht prop Dylan Rogers sent to the sin-bin for cynical play after hampering Cillian Willis at the back of a ruck, Nacewa kicked Leinster level.
Willis found a soft centre to Connacht's defence again, but Connacht clawed back territory and Niall O'Connor slotted his second penalty for a 6-3 scoreline.
A surging run from Nacewa got Leinster back on the attack and with Connacht caught flatfooted, Toner powered onto a pass from Willis and barged past final defender Matthew Jarvis for a 20th-minute try.
Nacewa could not add the extras and Connacht continued to force the issue in the second quarter. A long, inviting pass from Niall O'Connor put Ofisa ghosting inside David Kearney for a try which the creator converted.
With their confidence up, Connacht stretched their advantage to 12 minutes courtesy of a neatly-taken try from winger Tuohy. He gathered Henry Fa'afili's pass and slipped past Nacewa to score to the right of the posts with Niall O'Connor again kicking the conversion.
Leinster were much improved after the break.
They recovered their poise in the early stages of the second half and Nacewa landed two successive penalties to reduce the arrears to 20-14.
Tuohy roused Connacht with a spirited kick chase up the left wing, but he knocked on when attempting to regain possession just metres from the Leinster line.
Further indiscipline from Connacht allowed out-half Ian Madigan to send over a huge penalty in wet conditions.
With replacement prop Nathan White helping Leinster win a scrum against the head, Leo Auva'a powered off the base and the impressive Willis passed for Fitzgerald to slip over by the posts.
Nacewa converted and added a penalty as the visitors' challenge faded.
Connacht, struggling with injuries and illness in the squad, were unable to respond and a drop goal from Madigan closed out the scoring on the stroke of full-time.
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