The midfielder scored the only goal of a tense triumph over the Turkish champions, and the efficient manner in which United closed out the victory hinted at a growing maturity in a side whose inexperience was so ruthlessly exposed in European competition last season.
"It's vital we got off to a winning start after last year, really," Carrick told Sky Sports. "It was vital we won at home, especially against a good team - and they really are a very decent team.
"They kept possession very well. At times I thought we defended well, especially in the second half when we limited them to possession without really hurting us. That was a good sign for us.
"They counter-attacked on us quite well and were a bit of a threat, but we nullified that in the second half and overall we can be happy; we've played some good stuff and we've won the game."
Carrick showed commendable honesty and desire to hook home the only goal of the game despite being hauled down by goalkeeper Fernando Muslera, and the midfielder saw the funny side of his first Champions League goal in almost three years.
He laughed: "It's hard, you know, I don't get many so when the chance is there... obviously you can go down and stay down and there's a chance maybe of him getting sent off, but we've missed three penalties now so maybe taking the chance was the right one!"
The Reds did later earn a penalty when Rafael was tripped by Burak Yilmaz, but Nani subsequently became the third United player this season to miss from the spot - a sequence of events which Carrick concedes Sir Alex may have to address.