Ulster fell short of securing their own try bonus, which would have automatically brought them into the last eight ahead of next Saturday's game at Welford Road, and now need to beat the Tigers, who are expected to see off Treviso, to finish top of the group.
"No matter what happened here it was always going to come down to Welford Road and nothing's changed," Anscombe said after tries from Robbie Diack, Ruan Pienaar and John Afoa along with 12 points from the boot of Pienaar secured what was a hard-fought win.
"We're five from five and we're proud of that and no one else is. It wasn't ideal work but it's a win.
"We created a few chances but didn't quite do it as I said before the game if we got four points. If you have six fours you qualify and we're on track.
"At the end of the day a lot of people started expecting the bonus point but that wasn't showing enough respect for Montpellier.
"They came here to play some rugby and they did that and the more the game was unstructured it suited them and we didn't help our cause there.
"We had moments to put it to bed but we didn't quite do that and that's something to work on and improve and fix during the week."
Ulster only led 17-13 at half-time before pushing on to secure the four points and they then chased but failed to grab the maximum return from a bonus point try.
"They wanted to unsettle us and thought we didn't start too badly we didn't look after the ball and gave away a few penalties and our discipline turned us a bit," Anscombe added.
"Lazy penalties will allow teams to put us in the corner and accumulate some points and they had a class goal-kicker," he added about Eric Escande, who kicked 11 points to keep Montpellier, who had nothing to play for, in the chase.
"We know that next week they (the Tigers) will be more than happy to take these so we have to make sure our discipline is sharper and we play with confidence," the coach said.
"I said to the guys after the game it's very easy to beat yourself up over a performance and not be totally but you've still got to remember you're playing quality sides."
Ulster out-half Paddy Jackson - whose cross-kick created Ulster's opener for Diack - admitted that it was a difficult enough night for his side.
"At times it was a bit frustrating and we were playing off the back foot quite a lot which made it difficult," said the Ireland international.
"A couple of our tries came from kicks and they were out to spoil our party.
"We knew it was always going to be difficult and their defence hit us hard and they had a blitz defence so it's something we have to deal with in the future."
- Sports & Recreation
- Ruan Pienaar
- Robbie Diack