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."
Allardyce added: "All I can hope is that the panel, in the cold light of day, see it for what it is."