Irish frontman Daly thanked manager Craig Levein for giving him his chance, after he expected to sit the match out.
And United's first victory over Celtic in 10 years was an occasion to celebrate for the home team.
"When you go a goal down it's always going to be tough against Celtic," said Daly. "But we showed great character and there were two great balls from Swanny [Danny Swanson] from the corners and me and Dodsy managed to get our heads on them.
"I'm just happy to be back and to be playing games like that and scoring goals, it's great.
"It's been seven months (since I last played) and this week I've been off with flu as well. I didn't really expect to be involved today but when the gaffer put me in the squad I was delighted and hopefully I've repaid him there."
Dods had fouled Barry Robson to hand Celtic a penalty which Robson, the former United captain, calmly slotted into the net.
But defender Dods made amends with his late winner.
Dods told ESPN: "I was disappointed to give away the penalty but I was delighted when Daly scored to put us back in the game.
"It's been 10 years since we last beat them so we're delighted to get the win."
United moved level with third-placed Hibernian with their win, two points behind Celtic and with a game in hand.
"We've got tough games coming up but we're just trying to keep on Hibs' tails at the moment," Dods said.
"We're trying to get ourselves up to third."
Rather than chasing down the Old Firm, Dods stressed: "We're trying to get in front of Hibs first."