Hickies' try made amends for his second-minute blunder which resulted in Paddy Wallace scoring for Ulster, however he was back to his best when he barged through Neil Best's tackle to score with five minutes remaining.
Wallace's try got Ulster off to a great start but Contepomi landed two penalties penalties after five to reduce the arrears to 7-6.
Leinster got their first try on 21 minutes when flanker Ciaran Potts avoided a Wallace cover tackle to score in the left hand corner.
Contepomi's third penalty gave Leinster a 14-7 advantage into the break.
Contepomi and Humphreys exchanged two penalties apiece before Best went over in the 69th minute, with Humphrey's conversion making it all square with ten minutes left on the clock.
However Leinster, unbeaten at Donnybrook in the last 18 months, worked their way upfield to set up Hickie to score his 19th league try and join Derrick Lee and Matthew Watkins at the top of the scoring charts.
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