First, Nikora dinked over a well-taken drop goal and then a thrilling counter attack, sparked by Duffy and continued by some superb offloads, ended with second row Swift, the province's most-capped player, racing down the left touchline to score in the corner.
Nikora's conversion added to Connacht's sudden momentum at 17-3 and though Berne clawed back three points, another scintillating run by Duffy set the westerners up for their third try, cheekily scored by Nathan off a ruck in front of the posts.
Replacement Ian Keatley's conversion took Connacht's tally to 24. Leinster needed a break and they got it when Johnny O'Connor, also on as a replacement, was sin-binned for playing the ball on the ground.
Three minutes later, Leinster were awarded a penalty try - as the home front row gave way - and Berne's drop-kicked conversion closed the gap to 24-13.
But the tireless Muldoon stoked the fires for Connacht once again and when Cian Healy was pinged for not rolling away outside his 22, his old Belvedere school-mate Keatley converted the penalty to put Bradley's side out of sight.
Speaking after the game, Michael Cheika, who has named a youthful Leinster side for Friday's visit to Glasgow, said: "Connacht have shown pretty convincingly all season that they are a good side, even in the game up in Dublin when we were lucky to win.
"I don't think that they're a bottom of the league team and we know that they have a lot of quality because we have coached a few of them ourselves.
"We lacked an edge in defence. We had a lot of possession but we didn't use it properly, but in the second half we attacked the line better.
"We worked hard, but it's another game that we have lost here because we lacked the correct attitude.
"Brian (ODriscoll) finished the game and he's fine, but Girvan (Dempsey) suffered a neck injury and we just have to manage our resources now with a game coming up against Glasgow on Friday."
|Connacht Rugby Score Card|
|Leinster Rugby Score Card|