It took Wolves until the 98th minute to break the deadlock as Bright Enobakhare converted a Barry Douglas cross.
The hosts also had Norris to thank for some heroic goalkeeping to preserve Wolves’ clean sheet on the night.
Norris admitted it was a busy night for him but praised his team mates for going on to win the game.
He said, “Yeah, it was a fairly busy night for myself but I’m just here to do my job and it’s all I can do really.
“We’re only three games into the season for me but I’ll make sure that I’m absolutely ready for every opportunity in case I’m called upon.
“I can only pull on the shirt, do the best I can , get the clean sheets and have a real good go in this cup run.
“That’s how we’ve got to look at it - we’ve got as many games as possible in this cup, it’s only going to be good for me.
“Bristol gave us a really good test, I thought they were absolutely excellent - full credit to them.
“The main thing is that we’ve got through - a few extra minutes under our belts which we didn't really want but I thought the boys did really well to keep our composure, keep the ball moving well and ultimately to get the win in the end.
“I prefer the quiet games, I’ve always said that but when I’m called upon, I’m just pleased to make the saves.
“We’ve got three clean sheets from three games - I’m pleased with that! It’s just how far can we get in this competition - in all competitions, when the FA Cup starts I’m sure it will be the same. How far can we go?”
Wolves have now been drawn against Manchester City at the Etihad Stadium in the fourth round of the Carabao Cup.
Norris is delighted for Wolves to progress as he says it is providing players with a chance to show Nuno Espirito Santo what they can do.
“The more games we have in this cup, the more games I'll play and as a footballer, all you want to do is play games - the more the better really,” Norris said.
“It gives the players who aren't getting much of a look-in in the league a chance to go out and get a full 90 minutes.
“The young lads coming through are experiencing some first team football as well - it’s excellent for the whole squad really.
“The more games we have in the competition, the better and it leaves the door open for the players to give the gaffer a problem.”
Wolves tie against Manchester City will be played during the week commencing 23rd October - further details of the fixture as well as ticket news will be announced when the date is finalised.
