United, who beat Spurs on penalties in last season's Wembley final, were joined in the semi-finals by Aston Villa after the five-times winners beat Premier League stragglers Portsmouth 4-2 at Fratton Park.
Big-spending Manchester City host Arsenal while league leaders Chelsea travel to Blackburn Rovers for Wednesday's quarter-finals.
With the rain sweeping down, young Irish midfielder Gibson opened the scoring for United against the run of play in the 16th minute and doubled his tally 22 minutes later with a powerful shot to the top corner from 18 metres.
The scoreline flattered United's youthful-looking team, however, with Tottenham playing the better football for much of the first half as they sought their first win at Old Trafford in 20 years.
"We've scored against the run of play, Tottenham were I thought dominating the first 30 minutes of the match anyway," United manager Alex Ferguson told Sky Sports television. "They could have been in front by two or three opportunities.
"The second half, in fairness, we were much better. We are two games away from Wembley, so that's good...we will enjoy the semifinal and these lads will play again."
REDKNAPP STUNNED
Jermain Defoe tested United goalkeeper Tomasz Kuszczak in the seventh minute and had a great chance in the 21st when he found space at the back post only to be denied by the rock-steady Nemanja Vidic in defence.
Tottenham manager Harry Redknapp was stunned by his side's lack of passion.
"We were never going to get a better opportunity than we had tonight," he said. "The first half really, they (United) had two shots on goal and scored twice. There was no atmosphere, it was like a practice match.
"It was played at a lethargic pace, we didn't make a tackle in the first half. I've never been to Old Trafford and sensed there was no atmosphere. The thing never really got going, there was nothing happening."
Villa made a woeful start at Portsmouth, going behind in the 10th minute when Stilian Petrov tried to clear at the near post but sliced the ball into his own net.
Portsmouth, who lost 4-1 to United at the weekend, had little time to enjoy their lead with Emile Heskey equalising two minutes later.
Villa made it 2-1 in the 27th through James Milner and went 3-1 ahead in the 74th when England winger Stewart Downing headed in at the far post to score on his first start for the club.
Kanu pulled a goal back with a low shot in the 87th but Ashley Young restored Villa's lead two minutes later with a deflected shot high into the net.
"It was the sort of start that we didn't want," said Villa manager Martin O'Neill. "We were hoping that we could come out strongly and conceding an early goal was not part of the plan. But, if anything, it got us going."
