United arrived at the KC Stadium just over an hour before kick-off having made slow progress heading east and they looked like a side who had yet to warm to their task when the hosts went 2-0 up inside 13 minutes.
At that stage a surprise win for the Tigers was on the cards but Moyes' side fashioned a worthy response - Wayne Rooney's free-kick headed home by Chris Smalling before the England striker volleyed home a sumptuous equaliser in the 25th minute.
United might still have lost the game in the second half, Alex Bruce going particularly close for Hull when he rattled the woodwork with a header, but instead ran out 3-2 winners when James Chester - who had opened the scoring for Hull in the fourth minute - diverted an Ashley Young cross into his own net.
Reflecting on his team's sloppy start to proceedings, Moyes said: "It was a fantastic result for us after still being on the Humber Bridge for the first 20 minutes of the game.
"I think they only put one person on the pay toll when we were getting here and maybe that (delay) is the reason we never got over it and were a bit slow getting here.
"We were a bit rushed. That's not an excuse for how we started but it might be a reason for it.
"But we got back into it and deservedly so.
"There were a few scary moments and the end but we should have got ourselves a fourth goal and made sure.
"We got a bit of luck at different times but overall we warranted the win because of the way we went about it and kept going, even at 2-0 down."
Not for the first time this season Moyes had a word of special praise for Wayne Rooney, whose wonderful effort represented his 150th Premier League goal for the Red Devils.
As landmark moments go, it was a classy one from the England striker.
"Wayne played great today because he had to do two or three different jobs," said Moyes.
"They tell me that's his 150th Premier League goal today, and to score one like that, a nice volley, was really well taken.
"The pressure he put on the boy for the own-goal was great too, to get it out wide and get in the box."
Hull, who beat visiting Liverpool at the start of December, might have sensed another huge scalp when David Meyler put them two in front but Moyes never believed his men were out of it.
"I don't think anyone in the ground thought this game was over when we went 2-0 down or at any time," he said.
"If anything I think most people rubbed their hands and said 'hey, we're in for some game here today'."
Hull boss Steve Bruce, who has never beaten the club he captained as a player in 15 years of management, was left feeling for Chester.
The centre-half started his career as a United trainee, making a solitary appearance in the League Cup, and initially seemed on course for a golden day against his old employers.
"It could a been his day in many ways," said Bruce.
"He was at Man Utd from 10 or 11 and he'll have cherished playing against them and scoring after four minutes to give us the lead.
"Then he's headed one the wrong way, it happens. It'll be one he remembers for a long, long time.
"He's like all of us, disappointed. But he's a very good young players and he'll get over it."