Basel were within a minute of a shock away victory before Ashley Young's header made it 3-3 and rescued a point for United.
But Basel remain leaders of Group C with four points from two games while Sir Alex Ferguson's men have just two, and Carrick said the players had "no excuses" for throwing away a 2-0 lead.
Carrick said: "It's not the end of the world but it's not ideal and we've left ourselves a lot of hard work to do but we are capable of doing that.
"There are no excuses really - we said at half-time that the game wasn't over.
"It was a bit of a shock to find ourselves 3-2 down after being 2-0 up. But we've no excuses, we threw it away. It's not like us at all but I'm sure it's a one-off and that we will bounce back."
Carrick also paid tribute to Young, who was making his debut in the Champions League after signing from Aston Villa in the summer, for holding his nerve to head the equaliser.
"We expect it of ourselves to keep going to the end and that if we get a chance, we'll score," added Carrick.
"Thankfully Ashley put it away and we might even have nicked a win in the end when Berba [Dimitar Berbatov] had half a chance. But that was probably asking too much.
"Ashley has settled in quickly and bought into everything about the club and how we approach things. He's got bags of character, he's kept going and got his reward."
United boss Sir Alex Ferguson has expressed his concerns about United's lapses against Basel, and the defence also looked shaky against Stoke last weekend, but Carrick is confident the Premier League champions will get back on track at home to Norwich on Saturday.
"I've no doubt that we will be ready for that game," he said.
"I'm not saying it will be easy by any means but hopefully we can put in a good performance and get the win."
Danny Welbeck had scored two goals in 80 seconds to seemingly put United in control in the first half - a "bitter-sweet feeling" according to the 20-year-old England striker afterwards.
But Welbeck may not feature against Norwich after Ferguson revealed he had hoped to substitute the player against Basel when he tired in the second half - but decided to keep him on as United went on the hunt for an equaliser.
Ferguson said: "In the first half he did very well but in the second half he tired and wasn't really in the game.
"After they scored the second goal so quickly you have to keep as many players on the pitch who can score a goal so that is why he had to stay on."