Montpellier captain Fulgence Ouedraogo finished off a breakaway try midway through the first period at the Stade de la Mosson as the hosts led 13-6 at the break.
Francois Trinh-Duc extended Montpellier's lead early in the second half but in the final quarter Leinster finally began to wear down their opponents.
Second half replacement Sean Cronin barrelled over in the right corner with 14 minutes remaining. Despite Sexton's successful conversion, it looked like Leinster would have to make do with a losing bonus point.
But right at the death referee Dave Pearson awarded the visitors a penalty and Sexton brilliantly held his nerve amid a cacophony of boos to grab a share of the spoils.
Leinster were missing the injured Brian O'Driscoll, Cian Healy and Shane Horgan from the team that won last season's Heineken Cup final against Northampton, with Heinke van der Merwe, Fergus McFadden and Rob Kearney the men to replace them.
Summer signing Damian Browne, no stranger to this kind of environment after his year spent at Brive, partnered captain Leo Cullen in the second row in the absence of departed lock Nathan Hines, while Isaac Boss was preferred to Eoin Reddan at scrum half.
Montpellier, with just two wins from nine Top 14 fixtures so far this season, insisted during the week they were hoping to use this match as a catalyst for kick-starting their season.
They were able to recall flanker Ouedraogo and out-half Trinh-Duc for the first time since the Rugby World Cup and, having moved the game to the city's larger football stadium, they were keen to make an early impression.
There was a near-capacity crowd at the Stade de la Mosson and they were in full voice when Benoit Paillaugue kicked last year's Top 14 finalists in front after four minutes.
Sexton soon replied for Leinster though and he fired over a second penalty from distance after 16 minutes. It was a hugely physical encounter and Montpellier looked threatening but Leinster had been in a promising position when the game turned after 23 minutes.
With men out wide, Isa Nacewa tried a grubber kick through to the right corner only to be blocked by Montpellier's Argentinian full-back Lucas Amorosino.
The chase was on for the line then with McFadden looking the favourite to win it but when he did not manage to pick up the loose ball, Amorosino found the onrushing Ouedraogo who had the freedom of the Leinster 22 to slide home under the posts.
Paillaugue converted and when he struck home a long-range penalty, Montpellier suddenly found themselves 13-6 in front with half-time approaching.
McFadden was held up five metres short of the line as Leinster sought a quick response but only a last-ditch Luke Fitzgerald tackle prevented Geoffrey Doumayrou getting another try at the other end on the cusp of the interval.
It was intense, physical stuff with both number sevens, Sean O'Brien and the huge Georgian Mamuka Gorgodze, coming in for special treatment from the respective defences.
Shane Jennings and Eoin Reddan replaced Kevin McLaughlin and Isaac Boss early in the second half as the two-time champions started with renewed determination.
But the home side stretched their advantage to 16-6 through Trinh-Duc and Leinster may have thought it was not going to be their day when Sexton fired badly wide with a penalty of his own on the 22-metre line.
However, Joe Schmidt's men are not champions for nothing and just a couple of minutes later, Cronin finished off a sweeping Leinster move out wide, showing great pace and power.
Sexton was successful with an extremely tricky conversion attempt, meaning Leinster trailed 16-13. The momentum was firmly with Leinster now and they continued to batter the Montpellier rearguard. They were finally rewarded at the death when Sexton slotted over to salvage a hard-earned draw.
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