Glasgow edged ahead at half time thanks to a Chris Fusaro try converted by Stuart Hogg in the 39th minute, after Jimmy Goperth's penalty had given Leinster an early lead.
The Warriors then dominated the second half but, due largely to three separate kickers missing six shots at goal, they were dependent on a late try from Jerry Yanuyanutawa to give them the win after Goperth's second penalty cut their lead to just a point.
Italian side Zebre shocked Cardiff Blues by claiming their first Pro12 victory in 25 attempts, but the match ended in controversy with the visitors appearing to have an extra man on the field for a passage of play in the closing minute.
With replacement prop Luca Redolfini in the sin bin, Dario Chistolini was called back on to complete the front row for a scrum, without anyone making way for him until the vital phase was over.
That said, the visitors, inspired by New Zealand scrum-half Brendon Leonard, were well worth their 30-25 success, crossing for three tries to the Blues' second-minute effort, made by fly-half Rhys Patchell for centre Dafydd Hewitt, Leigh Halfpenny converting.
Halfpenny, returning to action for the first time since his Player of the Series contribution to the British and Irish Lions' triumph in Australia, supplied 20 points with the boot, but it was to no avail.
Munster were knocked off the top of the table as they were upset 29-19 by Benetton Treviso.
Ludovico Nitoglia had two tries while Mat Berquist kept the points flowing from his boot as Munster missed out on the chance to become the first side ever to begin a league season with maximum points from their opening three games.
Keith Earls and Stephen Archer both crossed Munster, but ill discipline cost them as Berquist amassed 19 points, scoring five penalties, while three different Munster players endured spells in the sin bin.
Newport Gwent Dragons continued their great start to the season by securing a 23-16 win in a titanic Welsh derby with the Scarlets at Rodney Parade.
Both sides played some excellent rugby throughout a fantastic match which saw Dragons prop Phil Price, on his 50th game for the side, and full-back Dan Evans cross for the home region while centre Scott Williams got a cracker in reply.
Ex-Wales wing Tom Prydie booted two long-range penalties, with fly-half Jason Tovey adding both home conversions and a penalty.
Wales outside-half Rhys Priestland booted three Scarlets penalties and converted Williams' try but, despite some intense pressure in the final throes of the game when a converted try would have drawn it, the Scarlets could not cross again.
Paddy Jackson plundered 13 second-half points to guide Ulster to their first win of the new campaign against Connacht.
Jackson put a disappointing first 40 minutes behind him as he converted his own 57th-minute try and kicked two penalties to decide this first Irish interprovincial derby of the season.
The sides scored a try apiece in the opening quarter hour, Ulster winger Michael Allen replying to a try from Dan Parks whose conversion separated them at the break - 7-5.
But Connacht were held scoreless for the remainder, with man-of-the-match Iain Henderson, who saw action in the second row and back row, and replacement Paul Marshall helping Mark Anscombe's men find an extra gear in their territorially dominant third quarter.
The Ospreys' unbeaten start to the season continued as they thrashed Edinburgh 44-10 at the Liberty Stadium.
The 2012 champions scored five tries in the bonus-point win as they made light work of a disappointing Edinburgh outfit who were second best throughout.
Richard Hibbard and Eli Walker both scored twice, with Jeff Hassler claiming the first try after just two minutes and Matthew Morgan completing the rout, while Dan Biggar added 12 points from the tee.