The Red Devils picked up an impressive away result in the Premier League, winning 3-1 at Tottenham on Sunday.
United had to battle back in the game, after a Heung-Min Son opener gave Spurs a 1-0 half time lead.
Maguire believes the team can improve, especially in how they start games.
"Well, I think it showed in the first 30 minutes that we played, we couldn't really get up to speed," Harry told reporters, per the club website.
"There was nothing in the game and we probably had a bit more control in the game, but we didn't play aggressive, we didn't play on the front foot, we didn't win enough second balls. After about, I think, probably about 30 minutes we dominated and got the result we deserved.
"I think sometimes when you play away in Europe the hardest part of the game is the start. I know it doesn't really make sense or add up, but when I've looked at our performances after Europe we need to start the games more aggressive, more on the front foot, more tempo.
"I think we started a bit slow and it's hard to get going. In the end, I think we were 2-1 up and we're trying to hold onto the lead and we score, I think the energy is there, the adrenaline. It's something that we do need to address when we come back from Europe, to start the game a lot quicker than we have been doing."
Asked about United's comebacks in the league, he added: "Yeah, first and foremost, it's incredible [that we keep coming back] and we need to stop conceding that first goal and make it a bit easier for our fans and the staff to watch.
"But we're giving them great entertainment; it's not like we're trying to do that. Like I say, it's something that shows great character and spirit. We have goals in this team and I think the three goals that we scored were high quality.
"So I think it's really important that we know that we can always get back into the game. I think we've showed that this year."