First-half tries by Sean O'Brien and Cian Healy put Leinster ahead, with two penalties from Ruan Pienaar keeping Ulster in touch at half time.
A penalty try for the defending champions set them on course to retaining their title, with Dan Tuohy's score turning out to be scant consolation for Ulster after Heinke van der Merwe and Sean Cronin crossed for Leinster.
O'Driscoll admitted Leinster set themselves a target of becoming one of the greatest sides in European history after they won their first title in 2009.
He told Sky Sports 3: "It will take a couple of days for it to properly to settle in but after we won one we talked about not being content with that and trying to create some sort of dynasty and something to be remembered by.
"We're going in the right direction towards doing that but I know this team will be hungry for more .
"That's why it's enjoyable, You're going in day-in and day-out because you know you're going to be pushed by the guy next to you. We played for one another out there and that really showed today.
"The energy from all 23 was phenomenal, it shows where we are as a squad. I only came in for the knockout stages. For the important part, but the hard graft was put in during the group stages."
Man-of-the-match O'Brien revealed the winning Leinster side are still looking to push on despite claiming back-to-back European titles.
He said: "We've worked hard in the last 10 months and today was they day where we had to for go it and thankfully we got the win.
"It's a strong squad and we really want to kick on. Every time we put on a blue shirt there's an energy around us.
"We trusted the team and we trusted the system. At times today there were lots of mistakes but winning is important and that's what we did today."
Man-of-the-match O'Brien revealed the winning Leinster side are still looking to push on despite claiming back-to-back European titles.
He said: "We've worked hard in the last 10 months and today was they day where we had to for go it and thankfully we got the win.
"It's a strong squad and we really want to kick on. Every time we put on a blue shirt there's an energy around us.
"We trusted the team and we trusted the system. At times today there were lots of mistakes but winning is important and that's what we did today."
Ulster coach Brian McLaughlin, who tasted defeat in his last game in charge of the Irish province, admitted Leinster deserved the trophy.
He said: "It was really only in the last 16 minutes that Leinster moved away as the game was very much in the balance.
"We fought back hard and scored a nice try but in the last 10 minutes Leinster showed what a quality side they are and how much they deserved the trophy
"I wouldn't do anything differently today at all. We knew it was always going to be difficult."
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