The England striker, making his third start of the season and long heralded as the Hammers' likely saviour by Allardyce, twice provided cushioned headers for Kevin Nolan to score before being sent off for an altercation with Swansea defender Chico Flores.
Carroll caused chaos in the visiting defence, but 14 minutes into the second half he received a straight red card from referee Howard Webb after appearing to brush the defender with his elbow.
"We will appeal it, yes," Allardyce said.
"We will hope that the appeal will work. Hopefully we don't lose Andy for three games.
"He's gutted. He's absolutely beside himself. We'll just have to go through the process and see what happens in midweek."
A three-match suspension would rule Carroll out for the rest of February and matches with Aston Villa, Norwich and Southampton.
Allardyce added: "All I can hope is that the panel, in the cold light of day, see it for what it is."
The Hammers boss blamed Flores for the sending off which came after Carroll and the defender had been involved in an aerial duel.
Allardyce recalled the incident which so greatly amused him during West Ham's October trip to Swansea.
Flores had gone to ground clutching his face and moaning following a challenge from Carlton Cole in the goalless draw, and Allardyce was later seen laughing in the Spaniard's face because of his "squealing".
Allardyce told BT Sport: "That was him squealing and centre-halves aren't supposed to squeal are they?"
Referring back to Saturday's game, he said: "If we're going to red card people for that then we might as well stop playing and I might as well retire. "He (Webb) can't be absolutely certain of what's happened, except that Chico Flores is over-exaggerating, which is generally the norm for him unfortunately."
The win was West Ham's first at home in the Premier League since the November 30 defeat of Fulham and came after a morale-boosting goalless draw at Chelsea, after which Jose Mourinho accused Allardyce of deploying 19th century tactics
"It (the sending off) hasn't cost us anything by the actions and the brilliant 21st century defending by the players," Allardyce added.
"And we got quality finishing today which is something we've been striving for on a consistent basis.
"It's a great result for us and one that's taken a little bit of pressure off."
Carroll's performance provided a major boost to Allardyce.
"His ability to hold the ball up and cause problems aerially is obviously his major strength," Allardyce said.
"He just needs to start scoring a goal or two, which is now his next challenge, in the hope we don't lose him.
"He's made two today for Kevin. Make another couple the next game and score one yourself."
Allardyce reserved special praise for Nolan.
"His contribution today all over the pitch was a proper captain's contribution," Allardyce said.
Swansea manager Michael Laudrup admitted the decision to dismiss Carroll was "harsh".
He said: "I think there was contact, but I think it was a bit harsh, contact like we see a lot of times in every game.
"The rules say if there's an intention it's enough. It's like if you come in for a tackle and you don't catch the player, but you could've and it could be a red card anyway."
Laudrup was yet to speak to Flores about the incident immediately after the match.
The Swansea boss lamented his side's failure to cope with Carroll.
"We really struggled with them physically," Laudrup added.
"It's a little disappointing that the first goal came from exactly the situation we analysed and talked about before the game.
"The second goal is disappointing as well."