The England international looked to have handed the reigning Premier League champions all three points at Old Trafford with a deflected strike 10 minutes from time, shortly after Robin van Persie had cancelled out Steve Sidwell's first-half opener.
There was to be a dramatic injury-time twist, however, as Darren Bent headed home a 94th-minute leveller for rock-bottom Fulham.
United dominated possession and had 31 shots on goal, leaving Carrick at a loss to explain how David Moyes' men only gained a point.
"It's a tough one to take, it's hard to explain how we've only come away with a point," he told the club's official website. "It was an incredible game, so one-sided.
"We played well for most of the game, we created opportunities but just couldn't score. When we finally got in front, we should have extended the lead with the chances we had, so it's tough to take after being hit like that at the end.
"We certainly had chances to do it (go further ahead), but even so, we were so in control that letting it slip like that is criminal.
"I haven't been involved in a game like that in my career. It's so frustrating because we should have scored more. We're very disappointed to concede twice, considering the amount of time they spent in our half, never mind our box."
The draw leaves United's hopes of qualifying for the UEFA Champions League via their league position in jeopardy as the club currently sit seventh, nine points of adrift of fourth-placed Liverpool.
United head to the Emirates Stadium to face Arsenal on Wednesday, with Arsene Wenger's men looking to quickly eradicate memories of Saturday's 5-1 humiliation at Anfield.
But Carrick is looking for United to bounce back quickly and inflict more misery on Arsenal.
He added: "We'll be ready (for Arsenal). We'll pick ourselves up and get back to training in the morning. It's one of those seasons - setback after setback - and we have just got to pick ourselves up.
"We have to believe that with performances, things are going to change. There's no better place to do that than on Wednesday night at Arsenal, and we still believe that we can go there and get the result we want."
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