Rashford had spent much of the previous two seasons on loan at Aston Villa and Barcelona after falling out of favour under former United manager Ruben Amorim.
Barcelona decided against taking up their option to sign the 28-year-old permanently, giving Rashford the chance to return to Old Trafford under Carrick.
He was one of four changes made by Carrick following United's disappointing 2-0 defeat to Hull City on the opening weekend of the Premier League season, and played an important role as the Red Devils produced a much-improved display against Ipswich.
Carrick was delighted to see Rashford back in the side and believes the England international has embraced his opportunity.
"He's added to the group for us, for sure, because it's pretty obvious," Carrick said.
"I think he's making the most of that for us and for him as well, to have the opportunity to be here. He's one of our own. He's living his dream, playing here now, and trying to be successful and win.
"At this club, there is always players that have come back sometimes, or players that maybe were going to leave, didn't leave. It's always about what you do next, and what your intentions are.
"He's been fantastic since he came back, I have to say. It was great to see him on the pitch today for us."
While Rashford's return was a major talking point, Bruno Fernandes once again proved to be United's key man as he scored a hat-trick to inspire the comeback victory.
Ipswich took the lead, but Fernandes equalised with a powerful finish on his left foot before adding a penalty and a close-range goal in the second half.
The Portuguese midfielder, who was named PFA Player of the Year for last season earlier this week, once again showed why Carrick sees him as one of the leaders of the United squad.
"He's got a lot of experience now to pass on to others and show them the way a little bit as well and lead by example," Carrick said.
"Sometimes you're looking towards certain players to give us that little bit of something else and he did that today particularly well. He offers us a lot."
Carrick, who was appointed United's permanent manager after leading the club to third place in the Premier League during an impressive interim spell last season, was particularly pleased with his side's response to both the Hull defeat and falling behind against Ipswich.
United's poor performance in their opening game had quickly dampened hopes that last season's momentum could carry into a title challenge, but Carrick felt the character shown against Ipswich was just as important as the five goals.
"That's probably the biggest thing to take," Carrick said.
"There's a lot of highlights and there's a lot of really good football, but that mentality was what we really relied on today."
Manchester United will look to build on their first win of the season when they travel to face Everton at the Hill Dickinson Stadium next Sunday.
