David Moyes' side ran riot in what was United's biggest away win in Europe since beating Shamrock Rovers 6-0 in September 1957 in the now defunct European Champions Clubs' Cup.
"This is probably the best performance of the season. For us to come away from home and score five goals is really special," said Moyes.
A volley from right-back Antonio Valencia and an own goal from Leverkusen centre-back Emir Spahic gave United a commanding early lead before defender Jonny Evans added a third with an hour gone.
England defender Chris Smalling added their fourth on 77 minutes after he met Wayne Rooney's chip at the back post.
Nani then latched on to a Ryan Giggs long ball for a fine solo goal for the fifth on 88 minutes with the Leverkusen defence in tatters.
Rooney proved he can create as well as finish as he had a hand in the first four goals in an outstanding performance.
It was his cross which Valencia converted and his free-kick which Spahic headed into his own net.
His shot led to Evans' goal and his deft chip beat Leverkusen's goalkeeper Bernd Leno and found Smalling unmarked for the fourth.
United remain top of Group A
The result keeps David Moyes' United top of Group A and three points ahead of second-placed Shakhtar Donetsk, who United host at Old Trafford in a fortnight.
Donetsk's 4-0 win at home to Real Sociedad means Leverkusen drop to third in the group and a point behind the Ukrainians.
Having offered little resistance against United, Leverkusen now need to win at Real Sociedad in their final game, and hope United beat Donetsk.
Leverkusen deservedly suffered what was their first home defeat in 14 games since losing 2-1 to Bayern Munich in the Bundesliga back in March.
"When you conceded five goals, there isn't much more to say," said Leverkusen striker Stefan Kiessling.
"We should never have allowed them so many chances to shoot."
Now unbeaten in 11 matches since September, United got their wish to be able to focus on improving their sixth place in the Premier League.
Leverkusen allowed United to dominate
Despite all their pre-match talk about not being intimidated, as they were for their 4-2 defeat at Old Trafford in September, Leverkusen allowed United to dominate.
United's opener came when Kagawa won the ball in midfield, squared to Giggs, who found Rooney on the left wing.
The England striker floated his pass over two Leverkusen defenders and Kagawa, but found Valencia at the back post on 22 minutes.
The Premier League side doubled their tally when Rooney's free-kick was headed into his own net by Spahic, under pressure from United right-back Chris Smalling on 30 minutes.
Leverkusenhad a brief passage of pressure just before the hour mark, but at the other end, Rooney slipped his marker and threatened a third goal.
It came moments later from a corner when Rooney forced Leno into a save and centre-back Evans collected the rebound, rounded the stranded Bayer goalkeeper and tapped home on 66 minutes.
The fourth came 13 minutes from time when Kagawa slid his pass through the Leverkusen defence for Rooney to chip over Leno and give Smalling the simplest of tap-ins.
With the defence all at sea, Nani met a deft long ball from Giggs, rounded Leno and fired home for a fine individual goal which summed up United's night.