Carroll, 23, was signed on a season-long loan from Liverpool last week after Allardyce spent most of the summer chasing his signature.
The England international made an impressive debut in a 3-0 win over Fulham on Saturday but his day was tainted when he suffered a hamstring injury in the second-half.
Allardyce has admitted Carroll will be his first-choice striker and reckons he can secure the club a spot in the top-10 on their return to the top flight.
"If Andy scores between ten and 15 goals, it is going to give us a great chance to be out of the relegation zone," Allardyce told the cl6ub's official website.
"If Andy gets 15 to 20 goals, we can get in the top ten because all we have to do there is to produce enough clean sheets to make sure we get points on the board with the goals that he scores.
"Andy's going to be our main man and you've got to get people supporting him. I thought the support (against Fulham) from Matt Taylor and Ricardo Vaz Te was superb alongside Momo Diame."
West Ham skipper Kevin Nolan, who opened the scoring in the first minute, played alongside Carroll during their time together at Newcastle and Allardyce is hoping the friends can rekindle their on-field understanding for the Hammers.
"You can look at what happened at Newcastle in the first year when Kevin Nolan could feed off Andy," he said.
"By the time Andy had been transferred to Liverpool, he had scored eleven goals and by the time Kevin had finished the season, he had scored 13 and they were really comfortable in the division."