Everton boss David Moyes held his first team meeting as United manager on Monday at Carrington, and Carrick believes Sir Alex's legacy of winning games late on will live on under the former Toffees manager.
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"You have seen that for many years here, the desire and enthusiasm to never give in.
"That is engrained in the players and hopefully we can carry that on."
The former West Ham and Tottenham midfielder admitted he was gutted to hear Sir Alex announce his retirement and thought United would end his final game in charge with another late winner in the 5-5 draw against West Brom.
"I was just waiting for a last-minute winner from Giggsy or Scholesy," he added.
"On a normal day we probably would have finished it off because we had it won twice at 3-0 and 5-2. We let it slip but, credit to West Brom, they kept going.
"We were disappointed because we wanted to win the game for the manager and for Scholesy.
"We wanted their last memory to be a win but, in the grand scheme of things, it probably doesn’t mean too much. At least it was exciting for everyone who watched the game."