Having been involved in a relegation scrap for much of the season, West Ham now hold a comfortable position in the Premier League table, 10 points clear of the drop zone in 11th.
A significant factor in their resurgence has been the return of Carroll, who joined from Liverpool on a permanent deal in the close season following a successful loan spell but then missed the first half of the campaign due to a foot injury.
Allardyce frequently highlighted the impact of injuries when his side were struggling, and is not surprised to have witnessed an upturn in fortunes with Carroll and other key players available once again.
West Ham had won six of their previous nine Premier League games prior to their loss at the hands of Liverpool, which saw them provide a stiff examination for Brendan Rodgers' in-form team.
"When they all got fit we knew we'd be a better team and a better squad," said Allardyce.
"At the end of the day, a lot of your team depends on how good your centre-forwards are and when they're available.
"Missing Andy Carroll for so long has always been hard.
"As good as Carlton Cole has done, particularly - he's scored six goals this year - Andy Carroll is one of our main investments, one of our best players who can upset defenders and can score goals.
"When you haven’t got him for so long that can have an effect on the team.
"You can play as well as you like between one end of the pitch and the other but if you haven't got anybody who can create or score goals then that becomes a problem for results."
