First Carroll headed a cross down for midfielder Nolan, who controlled the ball before driving home the opener.
He then nodded a corner across goal for Nolan to guide home with his head.
The England forward was harshly sent off after appearing to catch Chico Flores accidentally with his arm on 59 minutes, but West Ham held on to win.
Flores went to ground clutching his face and the hosts thought he had over-reacted, with replays showing contact looked unintentional as Carroll tried to untangle himself after an aerial challenge with the centre-back.
Despite their numerical disadvantage, the Hammers defended resiliently to comfortably keep the visitors at bay and secure just a second win from their past 11 home league games.
The Hammers remain in the Premier League's bottom three, although they moved to within two points of Swansea, who produced a lacklustre display and have taken just two points from their past six away matches.
Chelsea boss Jose Mourinho accused West Ham of playing "19th Century football" after a goalless draw at Stamford Bridge in midweek, but Sam Allardyce's men showed more attacking intent here.
And they were rewarded when they took the lead with the game's first shot on target.
From a diagonal left-wing cross, Carroll cushioned a header down for Nolan, who controlled the ball on his thigh before driving in a left-foot shot.
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