Carroll twice provided the assists for Nolan to score - doubling his tally for the season - in the Hammers' first home win in the Premier League since November 30.
Hammers manager Sam Allardyce confirmed after the 2-0 win that the club would appeal against Carroll's dismissal for an altercation with Swansea defender Chico Flores which leaves the England striker facing a three-match ban.
"We're absolutely delighted, it was a great victory after a well deserved point on Wednesday night," Nolan told whufc.com.
"It's just marred by an incident, but hopefully that'll get overturned and we can just get on with it and look forward to what's going to be another tough game away to Aston Villa.
"Andy was gutted because he wanted to get minutes too, which is important for him, and he's honest enough to say it's definitely not a red. You sort of know when someone's done something wrong and I sort of knew he hadn't.
"He was very surprised, it was difficult, but we showed tremendous character to grind the result out."
Carroll was long touted as the Hammers' saviour by Allardyce and his understanding with Nolan - the pair were also team-mates at Newcastle - was evident for all to see on the striker's third start of the season following a heel injury.
"Everyone's forgotten what it's like because we haven't been able to do it on a regular basis for a long time," Nolan added.
"We knew that the more and more he got his fitness up, we had that in us.
"We've been working hard on the training pitch together just to make sure that we get it right, and we reaped the rewards."
Defender Winston Reid returned as a substitute after being out since November following ankle surgery and Nolan believes things are looking up for the Hammers.
He added: "We have just got to make sure we keep a clean bill of health now and if we do you'll see performances like (against Swansea) throughout the rest of the season.
"Saturday proved that with a mostly-fit squad, we will get results. Hopefully now, touch wood, you'll see us moving onwards and upwards."
In a wide-open relegation battle, eight points separate the bottom 11 teams, with Swansea in mid-table but just two points better off than the Hammers, who occupy the final place in the relegation zone.
Michael Laudrup's men ended an eight-match winless sequence by beating Fulham before falling to defeat at Upton Park.
Laudrup keeps his emotions in check.
"For me it's always the same," he added.
"We cannot go up or down just because of one result, one way or another.
"Last year there were a lot of teams involved, maybe not 11, but seven or eight until the last five or six games.
"The only difference compared to last season is that there's not one or two teams already stuck down there, eight, 10 points behind the rest, like we saw with Reading and QPR.
"It makes it more - I don't like the word interesting, because we're part of it - but at least for the neutral spectator it's interesting."