Hooker Aaron Dundon from Seapoint got them up and running with a seventh-minute try - their first touchdown in this year's competition.
Tullamore scrum half Richie Hughes scampered over for another try, four minutes later, and Dundon crossed the hosts' whitewash again before the break for a 19-0 score-line.
36-5 losers against Ulster last weekend, Connacht played into the wind in the first half and their resilience was not rewarded with OLBC out-half Luke O'Donnell missing a penalty attempt.
With the elements behind them, the westerners dominated possession in the third quarter with their forwards stringing together some great mauls.
O'Donnell landed a penalty to get Connacht on the scoreboard and Corrib centre Stephen Conlisk rounded off a real team try, with both backs and forwards involved, to cut the gap to 19-8.
However, holes then began to appear in Connacht's defence and they wilted in the game's closing stages.
Ger Dowd's men leaked three more tries in the final half-hour with winger Aaron Carroll of Roscrea (55 minutes) and Leinster's influential centres - Boyne's Bevan Duffy (72) and Cill Dara clubman Jack Ryan (77) - the scorers.
Seapoint out-half Brian Keegan, who kicked six penalties in the win over Munster, landed three out of his six conversions in the Galway sunshine.
Leinster's final game of the championship is against Ulster at Bangor (kick-off 2.30pm) next Saturday. It will be the title decider as Leinster currently have nine points but Ulster are still in the hunt with six.
Ulster claimed a losing bonus point in their 6-3 defeat to Munster at Ballincollig on Saturday.
In a tightly contested game, dominated by the forwards, Cobh Pirates out-half Eamon Ronan kicked Munster to victory with two well-struck penalties, with Limavady's John Semple landing a single penalty for the visitors.
Munster's final game is at home to Connacht next Saturday, with Richmond RFC in Limerick the venue (kick-off 2.30pm).