Manchester United took the lead through Bryan Mbeumo in the first half, but Dominic Calvert-Lewin equalised after the break to leave Michael Carrick’s side with only one win from their opening seven Premier League matches.
Maguire felt United had prepared well for the contest but did not make enough of the opportunities available in midfield. Speaking to Sky Sports, he said the team needed to be more confident when playing under pressure.
“We worked on it this week.
“We wanted to match them up without the ball and go quite aggressive. It was a bit of a five without the ball and a bit of a four with the ball.
“Fluid in midfield. We knew we could get a lot of joy in the middle of the pitch with them only playing two in there. That's probably the area of the game where we could've done more.
“We could've been a bit braver with the ball and taken the ball under pressure. But we're in a circumstance where we are not as brave as we should be, and we have to get that back because to win games for Man Utd we have to have that bravery and composure.”
The defender was more positive about the side’s organisation and determination, while acknowledging that Leeds exposed a weakness United must address.
“You can't question the togetherness, the spirit, the fight. This is a tough place to come. Leeds have been flying in the league.
“We can be better. We have to show more quality with the ball. When you come to places like this, you can't be beaten on set plays and we conceded a set play which got them back in the game.
“Throughout we felt comfortable in our shape. They didn't create too many chances. To concede a set-piece after half-time is disappointing. We had to dig in. We got a foothold at about 70 minutes and probably looked the more likely to go and win it but a draw is probably fair.”
Maguire also addressed the attention surrounding teammate Lisandro Martínez, who had criticised senior figures at the club following the midweek Champions League defeat to Atlético Madrid.
Martínez subsequently apologised, with United accepting his apology before the Leeds fixture. Maguire said the matter had been put behind the squad.
“There's been a lot of noise around the club this week - as there is every week - but more so, this week. Lisandro's come out, he's apologised, we've moved on from it.”
United remain under pressure to turn improved spells into wins. Maguire’s assessment was that spirit alone would not be enough: the team must be more composed in possession and more reliable when defending set-pieces if they are to climb the table.