The two teams met in the pool stages and each side won at home. Their game in Lyon on May 13 will determine which team comes out on top, wins the cup - and earns an automatic ticket into next season's Champions Cup.
The French club's South African contingent earned them their ticket to their first European final as outside-half Demetri Catrakilis kicked 17 points and Bismarck du Plessis powered over for a try.
The Dragons, in their second successive semi-final, earned huge praise for their brave defensive display and outscored their hosts by two tries to one with a strong fightback in the final quarter, after falling 22 points behind with an hour played.
Montpellier dominated the opening exchanges and enjoyed most of the territory and possession in a first half spent softening up the brave Dragons defence.
The visitors had the first chance to open the scoring, but Dorian Jones missed with a sixth-minute penalty.
His opposite number, however, was in no mood to miss. Catrakilis hit the mark with four successive penalties in the first half to give his side a 12-0 interval lead.
The Dragons had the chance to shoot for goal with the last kick of the half, but Carl Meyer was wide this time.
Dragons coach Kingsley Jones admitted at half-time his side needed to sort out their scrum, which was wilting under the pressure of two Springbok World Cup winners in the Montpellier front row and a 42-stone second row pairing - but they continued to suffer at the set-piece in the second half.
A neat grubber kick to the home line by Meyer almost earned the Dragons a try in the opening few minutes of the second period, but Catrakilis added more points from his boot to stretch the lead to 15 points.
Then came the try that gave the Frenchmen an unassailable lead. Australian scrum-half Nic White, on as a replacement, spotted some space in the Dragons' 22 after a turnover on halfway - and his pinpoint kick earned his side a five metre line-out. Bismarck du Plessis found fellow Springbok Pierre Spies at the tail of the line and then joined the juggernaut to claim the try.
The Catrakilis conversion was a formality and that was the end of the home' side's scoring.
The Dragons, though, were only just getting started. Meyer set up a try for Hallam Amos and then the full-back crossed for a score to ensure his side could leave the Altrad Stadium with their heads held high.