At Carrow Road on Saturday, Moyes saw his side claim their sixth straight victory in all competitions as they moved up to sixth in the Premier League table.
Rooney had played the full 90 minutes and scored in the 3-2 win at Hull on Boxing Day but his groin problem prevented him from being involved again two days later.
Instead, his England colleague Danny Welbeck came off the bench to score the only goal of the game on 57 minutes.
Afterwards, Moyes explained that Rooney was likely to be back in action against Tottenham on New Year's Day.
"Wayne was missing because he's got an adductor injury in his groin," the United boss told Sky Sports. "He missed the Stoke game because of the same injury.
"I think if we hadn't got so many games and we could give him a couple of weeks just to recover properly, I think he'd be fine but I'd expect him to be OK for the next game against Tottenham."
At the post-match press conference, Moyes revealed Robin van Persie was also on the mend, saying: "Robin has started some light training and he might not be too far away as well.
"We have not had them as a partnership very often, and we are looking forward to having them together."
Moyes added he was not planning a major spending spree when the transfer window opens again.
He said: "We have a lot of midfield players out injured just now, but we won't be doing a lot of business in January because a lot of the players will not be available, so I don't expect it to be a busy January really."
Moyes admitted his side were second best in the first half on Saturday and took pleasure in the fact they managed to claim another win on their travels so soon after Thursday's triumph.
"To win back-to-back games in three days in the Premier League away from home is really difficult," he added.
"Some teams don't play until tomorrow, but we had two games in three days and I've got to say the squad coped really well, even though we've had a few injuries.
"I was delighted to get the result. I thought Norwich played really well at times today and we had to dig deep to make sure we got the victory."
On his decision to bring on Welbeck in place of Ryan Giggs at the break, Moyes said: "I just felt Norwich had grown into the game - we'd started very well, and Norwich had got better - and I felt we needed to alter things a little bit.
"I thought Danny was the difference in the game. He came on, he ran and made good opportunities for us. I thought he played ever so well."
United are just two points off the top four ahead of the rest of the weekend action, but Moyes is not getting carried away by their good form.
"It's OK, but we've got to do better and we've got to keep progressing and winning games, and try and keep on the tails of the teams above us," he said.
"We'd like to be closer, we're working towards that and we're hopefully going to get there. We're playing well enough just now, we're finding ways to win and I think that happened a lot last year."
Welbeck was also all smiles after the final whistle, saying: "It's been difficult going away twice in three days over the Christmas period, so to come with away with six points is really pleasing.
"We've got a lot of momentum, but the next game is the most important, and we're not getting carried away looking at the past.
"I was delighted to come on and get the winning goal. I felt like I really played my part in the second half.
"It's a squad game and it's a long season, so we've got to rotate. That's what's happened today and we got three crucial points."