The home side raced into a 13-0 lead after 20 minutes courtesy of an Isa Nacewa try and two Jonathan Sexton penalties.
Paddy Jackson cut the deficit with two penalties though and Craig Gilroy's unconverted try got the Ulstermen within two points at the break.
Jamie Heaslip and Sean Cronin both crossed the whitewash for Leinster in the second half though, meaning Gilroy's second try was in vain and Cullen's men reached a seventh final in the eight-year history of the play-offs.
Cullen praised Ulster's first-half comeback but said his side's intensity made the difference.
"We need to focus on the next challenge. There's no point in winning the semi and losing the final. There are a lot of really ambitious players who want to win silverware," said the former Ireland international.
"We raced into a pretty decent 13-0 lead at the start but then we lost our way a little bit after that and Ulster built a lot of pressure on us. We lost control of the game.
"We regrouped at half-time again and we showed good intent to carry. We backed ourselves with a penalty to go for a scrum so that was pleasing.
"Ulster battled back incredibly well and they have shown their threats in the last number of weeks.
"It was a performance of really good intensity from us though.
"We had to use a lot of players to get here - 56 to be exact. It's been a challenging season coming off the World Cup.
"It's been hard to get a foundation without 20 players. They are all reasons,excuses. We are now where we are though."
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