Emil Riis struck in the fifth minute of stoppage-time to earn a 2-2 draw with the Blades.
The home side looked on course for victory after substitute Sander Berge made it 2-1 in the 84th minute.
But last push forward saw North End equalise, a free-kick from Ryan Ledson turned across goal by Jordan Storey for Riis to control and fire home at the far post.
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United had taken an early lead in the first half through Morgan Gibbs-White before Daniel Johnson levelled in the 19th minute.
McAvoy said: "To get a point at the end was brilliant and I think we deserved it over the piece.
"Sheffield United are a good team, they move the ball well and in terms of possession stats they are top in the division.
"We spoke as a group and we felt the best way to play against them and try and win the game was to go toe-to-toe. We did that.
"At times they broke our press and when they do that they can cause you issues but I felt we were a threat every time we managed to get the ball forward.
"There were good opportunities at both ends, I thought it was a really open game.
"Daniel Iversen made a great save at 1-1 and then we got an opportunity with Emil just before Sheffield United scored their second.
"At 2-1 down I didn't think we deserved to lose it. We changed it, I took Pat Bauer off and went 4-4-2, a system we haven't played much to be honest.
"Credit to our boys, they kept going right until the death and I'm delighted for Emil that he nicked that goal.
"I'm delighted for the fans and everyone connected with Preston. These guys give me everything and tonight you could see they will keep going right until the end.
"That for me is magnificent as a coach, you can't ask for much more."
McAvoy has seen his side score late on before and likened this comeback to his first game in charge last season.
"It ranked pretty similar to the equaliser late on against Norwich, my first game," said the Scot.
"I always say to these lads to keep going, you never know when something might fall for you in the box.
"We had some really good chances in the second half, got into some good areas."
McAvoy reserved some praise for late hero Riis who has six goals for the campaign.
"Emil has come in and done really well. He's young, he's learning, he will make mistakes," said McAvoy.
"He knows that we have that trust and belief in him. He needs to keep that belief in himself.
"We tell him that if he misses one, another chance will come along. Credit to him."
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