The French outfit eventually clinched an 18-17 success in front of 10,000 spectators at Twickenham in a back and forth contest.
After Vincent Rattez had crossed for the Top 14 side, tries in the space of 13 minutes from Harry Wells and Jasper Wiese turned the game on its head for Tigers, but man of the match Johan Goosen got a crucial second score for Montpellier with 22 minutes left and it proved decisive.
“I think we have to be very proud of everyone at Leicester Tigers,” head coach Borthwick said.
“I think the supporters should be proud of the way Tom (Youngs) led the team and the way we attacked the game.
“Yes we got in a position to win and we didn’t so there is plenty to learn, but that can be a positive that we got so close and there is still growth. We are disappointed but still proud.”
After successive 11th-place finishes in the Gallagher Premiership, Tigers have been much improved since the arrival of former England captain Borthwick last summer.
In with a strong shout of qualifying for the Heineken Champions Cup and after going within a whisker of a first European title since 2002, the signs suggest the Mattioli Woods Welford Road club are on the way back to their past glory days.
With a number of players involved in a showpiece fixture at a relatively young age, the Leicester faithful will hope it is the first of many finals for this group.
Borthwick added: “One certain positive out of this game is that we’ve played against Montpellier, who are an outstanding team, huge budget, players star-studded throughout and we came very close to beating them.
“At the same point in time, we’re sat here going ‘we could be so much better’ and when you look at Freddie Steward, you look at Dan Kelly, you look at Cam Henderson, who came on, Tommy Reffell, who’s only 22 years old, you’ve got a number of players here that have played less than 20 games for Leicester Tigers which is astronomical.
“So, there’s a lot of potential for growth for us and that in itself is positive.”
While it was disappointing for one of England’s biggest clubs, one person from the country was able to celebrate in the shape of Alex Lozowski who started at fly-half for Montpellier.
On loan from Saracens, it has been an up-and-down campaign for the five-cap England international but he was pleased to give their loyal supporters some reward for a turbulent season.
Lozowski told BT Sport: “It’s really special. To think where we have come from, we struggled at the start in the Top14 and lost a lot of tight games.
“But to finish so strongly and now win a final, it’s amazing.”