The forward, back at Old Trafford after a season on loan at Barcelona, was not part of the matchday squad for Wednesday's friendly but travelled to Dublin to support his team-mates as Manchester United beat their old rivals 5-4 on penalties following a 1-1 draw. Carrick, speaking to ITV, said Rashford's reintegration since returning to training two or three days earlier had been seamless.
"He's our player, and he's come back great," Carrick said. "I've known Marcus a long time, and he gives us something a little bit different, and he's been in great spirit since he's been back. He's been back two or three days, trained, and it's just pretty normal, to be honest. We're looking forward to the start of the season."
On the pitch, Joshua Zirkzee gave United the lead in the 16th minute, finishing first time after a driving run and pass from Bryan Mbeumo, before Brenden Aaronson levelled for Leeds just before the half-hour mark, finishing off good work from former United winger Daniel James.
Bruno Fernandes struck the crossbar late on, and the match ended in bizarre fashion when several young fans ran onto the pitch to take selfies with the Manchester United captain, delaying the restart until referee Neil Doyle blew for full time. Noussair Mazraoui scored the decisive spot-kick in the shootout after Fernandes missed his own attempt and Leeds' James Justin struck the bar, sealing a 5-4 win in front of a rare capacity crowd at the home of Gaelic games.
Carrick, speaking to club media, said the occasion had exceeded his expectations.
"I really enjoyed the whole night. It felt like a proper game," he said. "The fans, I have to say, have been incredible since we set foot over here.
"The atmosphere, with that many turning up supporting us - we're the lucky ones to be involved in that."
The fixture again gave minutes to some of United's younger players, with Dermot Mee replacing Senne Lammens in goal and saving a penalty in the shootout, Jack Fletcher converting from the spot, and Dan Armer and Tynan Thompson also involved.
"The young boys did ever so well, each of them individually," Carrick said. "It was a tough game as they are a good team. You can see the changes Leeds made, the two XIs, are virtually both as strong as each other. It was a good challenge for us and nice to win, of course. Lots of positives."
United close their pre-season programme on Saturday against Ruben Amorim's AC Milan in Wroclaw, Poland, the final test before the Premier League campaign begins, with Carrick able to reflect on a sold-out away friendly that gave several fringe players a valuable run-out and offered an early glimpse of a settled, smiling Rashford back among his team-mates.