Carroll watched his Olympiakos team-mates inflict a damaging 2-0 Champions League defeat on his old club last week.
And he believes the players have been shirking responsibility since the retirement of Sir Alex Ferguson.
The 36-year-old, speaking to Sky Sports in Nicosia ahead of his Northern Ireland side's friendly against Cyprus, said: "(The first-leg win) was a shock for a lot of press people but the players knew themselves they had a chance.
"We watched a lot of Premier League games this season and knew United had not been performing. On the night when I was watching it live from the stands, I've never seen a Manchester United team like it.
"They've got good players but they had no hunger, no fight in the team so it was a shock in that way but not a shock the way we played. The goals we got were well deserved.
"At the end of the day you can be a good player but you have to give 100 per cent and give everything.
"For me Manchester United didn't give that and looked like they had no hunger and fight for the game.
"I don't know if they thought they would get an easy game but no professional should go into a football match and think that."
United have struggled for form since Moyes took the reins in the summer and Carroll, who spent four years at Old Trafford between 2001 and 2005, reckons the departure of a man in charge for 26 years has precipitated a change in attitude.
"They might have slacked off since Sir Alex left, it's football," he added.
"People have their routines. You can't do that in the Premier League. It's a big league with top and bottom teams fighting for points and you have to be prepared every game."
United will seek to keep their quarter-final hopes alive at Old Trafford on 19 March and Carroll admitted his team-mates expect a tough second leg.
"We're happy and everyone in Greece is happy because it was a fantastic result for us but the game isn't over yet," he said.
"We know United are a great team and know what respect to give them. They're going to be a different team at Old Trafford.
"It's only half-time but the good thing for us is that we haven't let an away goal in so it's a good result for us keeping a clean sheet and a lot of pressure on Manchester United.
"Hopefully we can keep them out, zero-zero at half-time and the fans will start getting on their backs. You never know in football."
Carroll's comments were echoed by another former United goalkeeper - Peter Schmeichel, who feels a number of players have not pulled their weight this season.
Schmeichel spent eight seasons at United during the 1990s, winning five Premier League titles and was also a member of the squad that clinched the Treble in 1999.
Talking to Sky Sports News, he said: "I think David is a great choice.
"He's shown he's got the pedigree, he's got the ability, he can stay in a place for a very long time - he showed that at Everton.
"At United, we want our managers to stay a long time but this year has been really tough on him.
"He'll learn a lot about himself, everyone will learn a lot about him and he will come out a lot stronger.
"I think he needs to sign a few players - how many I don't know!
"But I think, equally important, he needs to offload a lot of players. There's a lot of players there I'm very disappointed with.
"To be a Manchester United player, that carries a responsibility. You fight, never give in, even on days when it's not happening for you, you still have to try.
"But I've seen, throughout the course of this season, some players who haven't done that and this disappoints me.
"I think he will have learned who he can trust and who he can't trust. And this summer, I think you'll see a lot of players leave the club."