Keane also claimed manager David Moyes would have been "shocked at the lack of quality" in the team he inherited from Sir Alex Ferguson last summer, and ridiculed the theory that the Premier League is the strongest in the world.
Speaking on ITV, Keane reacted sharply to a post-match interview from Carrick, who succeeded him in the United midfield in the mid-2000s, in which his demeanour appeared jolted by what he had experienced.
"That interview was just like the performance: flat. He should say a bit more, have a bit more urgency even in his interview," Keane said.
"That just reflected United's performance tonight: flat, with no urgency.
"They keep saying, 'Ah, well, next game, next game'. For some of them there won't be another game for them. That's the reality."
Keane was an onfield motivator in his playing days but he fears there is insufficient quality in the current United squad to make a serious impact in European football.
"I don't think any of us could have predicted that," he said after United's miserable surrender in the Athens district of Piraeus.
"They (United) just never turned up.
"They (Olympiacos) were very, very good and technically they were better than United tonight."
Pressed on where United's problems this season have arisen from, Keane said: "There's a number of things: new manager, new ideas. United have missed out on some players over the past two years.
"There's some players playing tonight for Man United who just aren't showing enough quality."
He added: "I think United need six top quality players.
"I would think David Moyes has been shocked. I think when he went into United in the summer he probably looked at the players and was expecting great things.
"Privately he's probably been shocked at the lack of quality that he's working with."
United, Manchester City and Arsenal have all now suffered 2-0 first-leg defeats in the last 16, with Chelsea looking to avoid suffering such a disappointing outcome when they face Galatasaray on Wednesday.
Keane claims expectation of Premier League success in Europe is misguided.
"We've been brainwashed by this 'Premier League is the best league in the world' (talk)," Keane said.
"Nonsense. It's the best brand, but we've seen over the past few weeks with a lot of the English teams struggling that these teams have fallen behind the top teams in Europe, and United in particular."