Grayson paid tribute to Beckford for his nerveless strike.
He said: "It was good ball from Jonathan Howson and Jermaine got in there and nine times out of 10 he finishes them off and that's what he did.
"He gave us an opportunity to go on and win the game from there and with a bit of luck that's what we did. Full credit to the players who were outstanding today but it's what they've been like all season."
Referring to Brian Flynn being the last man to score the winner for Leeds at Old Trafford in 1981, Grayson added: "I saw Flynny the other week and he reminded me he was the last player to score the winner at Old Trafford.
"I told the players before the game this was the opportunity to be a new hero. I'm sure we'll be celebrating in Leeds over the next few days and the players might have tomorrow off then start ready for the Wycombe game at the weekend."
Leeds captain Richard Naylor hailed his side's heroic win, and admitted: "It was a long time coming."
Naylor told ITV1: "We've had to wait a long time since we last won here and I'm really proud of the lads and the supporters.
"I thought we had a few chances to make it two late on. They're always going to get chances at Old Trafford, you wouldn't expect anything else but we dug in when we had to."