Flyhalf Jonny Sexton's accurate boot, five inspired minutes by Rob Kearney and resolute defence helped the twice winners to victory, though Wesley Fofana's last-ditch stretch for the line and ecstatic celebration left Leinster fans biting their nails.
With just four points separating the sides, France international Fofana wheeled away in delight after scoring what he thought was the match-winning try with seconds remaining on the clock.
But replays showed he failed to ground the ball properly as he arched his back to touch it down.
"We were saying to ourselves that we would manage to make them fall apart. But Wesley lacked a few centimetres to score his try," Clermont captain Aurelien Rougerie said.
Relieved Leinster coach Joe Schmidt was full of praise for Clermont's spirit.
"They were a big threat until the end and in the final minutes it was very tight. Clermont were whole-hearted and I think we can be proud of having won such a clash," he said.
Clermont's ferocious late surge threatened to derail Leinster's bid to become only the second team after Leicester, in 2002, to defend the trophy since the competition started in 1995-96.
After a more sedate first half and with the French club leading 12-6, Ireland fullback Kearney sliced through the Clermont defence after 41 minutes to feed prop Cian Healy for a try.
"Their try helped them get the upper hand both physically and mentally in the second half," Rougerie said.
Minutes later Kearney caught a defensive clearance near the halfway line, took a few steps and launched a towering drop goal straight between the posts to stun the loud French support.
A fifth Brock James penalty for Clermont, playing their first semi-final, reduced the arrears to a solitary point but a 63rd-minute penalty by Sexton proved to be the game's final score.
Leinster will attempt to win their third trophy in four years when they meet Ulster at Twickenham on May 19.
"We did not see yesterday's game but we know what Ulster are capable of. It will be a tough match," said Leinster forward Leo Cullen.
With Ireland guaranteed the winners of this year's trophy Connacht will be assured of a place in next year's competition.