Adam Byrne and debutant Barry Daly both crossed for the home side at the RDS Arena with Isa Nacewa adding 14 points with the boots while Shane Delahunt got a consolation Connacht try.
The win stretches the Leinstermen's winning streak at home to 19 matches and saw them avenge the Guinness PRO12 final defeat to Connacht last year.
Connacht enjoyed the better of the early stages and took the lead when Craig Ronaldson successfully a converted a penalty after Cian Healy was penalised.
Ronaldson doubled the visitors lead after 13 minutes with a second penalty from the right hand side just outside the 22.
Leinster finally managed to get themselves onto the scoreboard after 23 minutes as the New Zealand born Isa Nacewa knocked over a straight forward kick from in front of the posts.
Flanker Sean O'Brien was then shown a yellow card for an illegal clear out at a ruck but it didn't prevent Nacewa levelling the scores after 32 minutes.
With O'Brien seconds away from returning to the field, it was Connacht's turn to be temporarily reduced to 14 men as outside-centre Peter Robb was given his marching orders.
Robb was stretching for an interception but only managed to knock on and referee John Lacey adjudged it to have been deliberate.
And it was the home side that took the lead into the half time break as Nacewa knocked over the resulting penalty with 38 minutes on the clock.
Connacht piled on the pressure as the clock ticked past 40 minutes but a knock on just metres out from the Leinster line saw their opportunity go begging.
The crucial score of the game came ten minutes after the break as Byrne scored his third try in Leinster colours.
The home side had a five metre scrum and after a few tight phases, hooker Jack Conan's good hands found Byrne in space with the winger cleverly waiting for the bounce before crossing.
Nacewa missed the conversion but had his fourth successful penalty of the afternoon just before the hour mark.
Leinster sealed the victory with five minutes left on the clock, as winger Barry Daly sealed a dream debut with try from close range.
Shane Delahunt added a late consolation try with an excellent line to break through the Leinster defence which had been impregnable up until the last minute of the game.