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The home side mustered 14 shots on goal compared to the visitors' six in a dull affair which failed to live up to the pre-match hype, with Marcus Rashford squandering the best chances for Ole Gunnar Solskjaer's side.
Rashford only had Mendy to beat after being played through by Juan Mata midway through the first half, but he saw his effort kept out by a strong right leg from the Senegalese shot-stopper.
The Red Devils striker was also denied in the dying moments of the match as he curled a strike towards the far corner of the net from just outside the box, only for Mendy to fly across his goal and parry the ball out for a corner.
Chelsea were unfortunate not to be awarded a penalty shortly before the interval as replays showed Harry Maguire had wrapped his arms around Cesar Azpilicueta while defending a free-kick, but Frank Lampard's men were otherwise unable to create any opportunities of note in the final third.
Shaw felt United would have secured all three points were it not for Mendy's brilliance between the sticks, as he told the club's official website:“I feel like we created the better chances.
"Their keeper made wonderful saves at times. I was right behind it Rashford’s shot and, like I said, their keeper was very good today. He made a couple of very good saves that kept them in the game.
“We defended much better than we have been, especially against a very good team like Chelsea who have high-quality players all around the pitch.
“I feel like they didn’t really create much, and we did well in that aspect - but higher up the pitch, we should have maybe grabbed a goal, or maybe even two, and taken the three points.”
Asked to address suggestions of both teams being happy to play for a point, Shaw responded:“Yeah, they were saying up there on the TV that neither team wanted to lose.
“You could see that with maybe the way they turned up with that formation – they haven’t played it a back five this season, so it was the first time. So I think maybe we felt that from them, but we always want to win every game.
"Of course it was important we didn’t lose today, and we’ll take one point rather than zero, but also we’re disappointed in the back of our minds that we didn’t take three.”
Shaw also expressed his belief thatthe Red Devils are beginning to hit their stride after a slow start to the campaign, beforedefendingSolskjaer's decision to rotate his starting line up following Tuesday's 2-1 win over Paris Saint-Germain in the Champions League.
“When we’re on the pitch now, it feels different to what it was before at the start of 2020/21. Before, we made a lot of sloppy, silly mistakes – individual mistakes – and I think we’ve learned from that and we’ve worked hard in training," the Red Devils defender added.
“It’s very good for the manager to maybe have those changes that he can make, but also very good for the team, because everyone wants to play. Everyone’s pushing. The ones on the bench want to be in the starting XI, so it’s good to have that healthy competition that is going to push the lads.