Carroll and De Gea backed by their managers
Liverpool striker Carroll, who has struggled since a big move from Newcastle United a year ago, did not find the net but used his physical prowess in the area as Daniel Agger headed in the first and then set up Dirk Kuyt for the late winner.
A contender last year for England's Euro 2012 squad, the 23-year-old Carroll has dropped well down the pecking order and has been in and out of the Liverpool side despite fellow forward Luis Suarez serving a long ban.
His team-mates have had difficulty refining their passing game to accommodate a tall front-man but boss Kenny Dalglish thinks Liverpool may finally have cracked it.
"Everybody knows Andy's prowess in the air, we've tried our best to develop that further and we'll also work and be patient with him," Dalglish told ITV Sport.
"His commitment was superb today so as long as you've got the right attitude and the right commitment we've got a chance of working with it."
Goalkeeper De Gea arrived at United from Atletico Madrid in the close-season after rave reviews but the Spaniard has not got to grips with the physicality of the Premier League and has recently been left out in favour of Anders Lindegaard.
The always nervous 21-year-old was crowded out in the box for Agger's header from a corner and could have made a better attempt as Kuyt thrashed the ball into his near post.
Sir Alex Ferguson, who has struggled to recruit top goalkeepers over the last 25 years with only Peter Schmeichel and Edwin van der Sar being big successes, put the blame for the first goal on his defence and not his 1.93 metres keeper.
"We didn't do well. [There were] too many defenders in and around the goalkeeper for the first one and the second one is just one of those lucky breaks you get," he said.
"I'm pleased with my players, they performed very well and dominated the game."