The game was Liverpool's first at Anfield since the publication of the Hillsborough Independent Panel's report, which exonerated fans in relation to the 1989 disaster.
The hosts played 51 minutes with 10 men after Jonjo Shelvey's sending-off but still managed to take the lead through Steven Gerrard's volley seconds after the interval.
However, Rafael equalised for United soon after while Robin van Persie ensured it was Sir Alex Ferguson's men leaving with all three points when he slotted away a late penalty after Glen Johnson had brought down Antonio Valencia.
Veteran midfielder Giggs, however, felt the visitors were far from their best.
"Over the last three or four years we've come here and played a lot better and not got anything - that's how it goes sometimes," Giggs told Sky Sports 1.
"It's always tough against a team like Liverpool who keep the ball well.
"Even if they go down to 10 men they don't change, they keep the ball, and that's always tough."
Giggs admitted, however, that it was his side's equaliser rather than Shelvey's dismissal that finally injected some belief into the United players.
"The crowd got up and we didn't play well, we didn't keep the ball well enough, so it was great to see Rafa's goal go in - that probably lifted us," he said.
Giggs and Gerrard released 96 red balloons in memory of the Hillsborough victims before the match as part of a number of tributes, but the former Wales international did not feel the emotion of the occasion got to his team-mates.
"No, I don't think so," he said. "It's just a tough place to come and Liverpool obviously wanted to put on a performance because of the occasion, and they did that."
Ferguson echoed his captain's view that United had been far from at their best.
"A win's a win, so we've got to be pleased with the result, but not the performance," he told Sky Sports 1.
Reflecting on Shelvey's red card, awarded after the Liverpool midfielder had gone in recklessly on United defender Jonny Evans, Ferguson added: "I thought it was a clear red card.
"I don't think he went for the ball - Jonny Evans went for the ball and got the ball - and the boy's gone in really dangerously and I don't think there's any other decision the referee could have given."
Ferguson and a fired-up Shelvey exchanged heated words as the player left the pitch - something that left the veteran manager unimpressed.
"He was trying to blame me, I suppose," he said.
"Once he looks at it he can apologise if he likes. I think he should do that. He's responsible for his own actions, I think there's no doubt about that."
Liverpool manager Brendan Rodgers predictably had a different view of the red card decision, and was unhappy with the award of the late penalty while also feeling the hosts were unlucky not to get one of their own.
"I thought the players were heroic in terms of performance and the spirit," he said. "They were brilliant and didn't get what they deserved.
"Jonjo Shelvey, if he gets sent off then Jonny Evans has to go as well. I think both players' feet are off the floor.
"I'm sure it's never a penalty, and then obviously Luis Suarez goes up the other end and gets a toe to the ball and the defender doesn't touch the ball, and he goes down and it looks a penalty.
"That was out of our control."