The Ulster men recovered from a scrappy first half showing to overpower the 14-man French opposition, and secure a meeting away to Clermont Auvergne in the next round.
The game was turned on its head early in the second half when Montpellier captain Paul Willemse was sent off, and Ulster took ruthless advantage of a French side that lost their discipline and received three further yellow cards.
It was Willemse who powered over to give his side the lead with 10 minutes gone, before Eymeri increased their advantage after 25 minutes, with Foursans converting both scores.
After 28 minutes, Rob Baloucoune stepped in from his right wing to break the line and use his searing pace to get in behind the defence. Although he was hauled down short of the line, Will Addison was on hand to pounce and dot down in reply.
Cooney converted to begin what was a typically strong day off the tee, before Foursans kicked a penalty for the hosts on the stroke of half time.
HT Montpellier 17-7 Ulster
Just two minutes into the second half, Willemse was dismissed for a head clash as he hit the ruck, striking Will Addison, and the game turned entirely.
Barely a minute later, Ulster took advantage and scored their second try, sweeping the ball through the hands of the backs before Baloucoune darted into the corner, with Cooney’s kick rattling the post.
Richie Murphy had loaded the bench with a 6-2 split, and in the absence of Willemse it was the Ulster pack that started to batter their way to the Montpellier line.
Nick Timoney was held up, but the pressure soon told and replacement prop Scott Wilson barged over from close range to pull his home province level, before Cooney kicked Ulster into the lead for the first time, making it 19-17.
Two further moments of madness followed for the hosts with Verhaeghe and Karkadze both binned within the space of 60 seconds to send Montpellier down to 12 men.
The numbers were put to use immediately, and David McCann was on hand at the back of the maul to dot down and extend the Ulster lead, as Cooney nailed another tough kick to make it 26-17 going into the final 10 minutes.
Scrum-half Cooney then made a key defensive interception before Stewart Moore launched a missile of a 50:22 to get Ulster deep into the French side’s territory.
With 78 minutes played, the backs made use of the superior numbers to send Stu McCloskey clear down the blindside, and the big inside centre smashed away a would-be tackler to ground the ball for a fifth try, with Cooney deadly from the tee.
As the clock ticked into the red, Jake Flannery broke into acres of space and released McCloskey before receiving the return pass. Flannery was about to find Moore for a certain try before Ngandebe deliberately knocked it on.
His yellow card came alongside a penalty try to round off a superb second half showing from the men in Saffron which ensured their European campaign continues.
Scorers:
Montpellier
Try: Willemse, Eymeri
Con: Foursans (2)
Pen: Foursans
Yellow: Verhaeghe; Karkadze; Ngandebe
Red: Willemse
Ulster
Try: Addison; Baloucoune; Wilson; McCann; McCloskey; Penalty Try
Con: Cooney (4)
Teams:
Montpellier
(15-9) A de Nardi; G Ngandebe; G Bridge; A Cadot; B Lam; L Foursans; A Eymeri;
(1-8) G Fichten; C Tolofua; H Williams; F Verhaeghe; P Willemse (C); A Bécognée; C Doumenc; S Simmonds.
Replacements: V Karkadze; E Forletta; B Erdocio; L Macharashvili; T Duguid; Y Camara; L Carbonel; M Dakuwaqa; P Lucas.
Ulster
(15-9) Will Addison; Rob Baloucoune; James Hume; Stuart McCloskey; Stewart Moore; Nathan Doak; John Cooney;<br>
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(1-8) Steven Kitshoff; Rob Herring; Tom O’Toole; Alan O’Connor; Iain Henderson (C); Matty Rea; David McCann; Nick Timoney.<br>
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Replacements: John Andrew; Andy Warwick; Scott Wilson; Harry Sheridan; Cormac Izuchukwu; D Shanahan; Jake Flannery; Dave Ewers.