Carroll, who has just returned to action following a long injury lay-off, made a brace of assists for Nolan before being dismissed in the second half at Upton Park.
The striker was sent off after clashing with Swansea defender Chico Flores and now faces a three-game ban.
And Nolan, skipper of the Hammers, said: "We're absolutely delighted, it was a great victory after a well-deserved point on Wednesday night.
"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."