The three goals they scored in their opening Group F clash against Celtic at Old Trafford equalled their total tally for the six group games in last season's competition.
But, while United were irresistable in attack, their defending left much to be desired with England's Rio Ferdinand decidely shaky in the first half and his international team-mate Wes Brown just as dodgy in the second.
Celtic tore at United like men possessed, but Sir Alex Ferguson's side had that bit more class with Louis Saha magnificent in attack and Paul Scholes driving them on tirelessley in midfield.
Saha was twice close to scoring in the opening 20 minutes, heading Gary Neville's cross inches wide of a post and then blazing a shot narrowly over the Celtic bar from 12 yards. But it was Celtic who took the lead against the run of play in the 21st minute.
Goalkeeper Artur Boruc's long kick downfield looked easy for Ferdinand to clear, but the England man made a hash of chesting it down and was robbed by Jan Vennegoor of Hesselink.
The Celtic striker, with a name more befitting Dutch ability than football, coolly pounced to rob Ferdinand, and take the ball on 10 yards into the United box and score with a crisp left-foot shot which goalkeeper Edwin Van der Sar could only help into the net.
Celtic's lead lasted only nine minutes, although the Glasgow side will feel hard done by with United's equaliser.
Boruc seemed to get only the slightest touch on Ryan Giggs as the Welshman ran clear into the box and rounded the Celtic goalkeeper.
But referee Lubos Michel pointed to the spot and Saha stepped up to drive his left-foot shot to Boruc's right beyond his dispairing dive. Giggs left the field injured and was replaced by Ole Gunnar Solskjaer.
Vennegoor of Hesselink placed a free header at Van der Sar soon after, but it was United who took the lead in the 40th minute.
Michael Carrick won his 50-50 challenge with Thomas Gravesen in the centre circle and found Scholes roaming free.
Scholes produced a glorious pass into the Celtic box where Saha held off Mark Wilson's challenge before scoring with a low left-foot shot from 10 yards.
Back came Celtic as Brown brought down Jiri Jarosik eight yards outside the United box on the right. Shunsuke Nakamura stepped up to curl a glorious left-foot free-kick over the defensive wall and beyond Van der Sar into the roof of the net.
United regained the lead two minutes after the interval. Scholes released Saha into the Celtic box and, although Boruc made a superb save from the Frenchman, the ball flew across goal for Solskjaer to stab home his left-foot shot six yards from the far post. It was the Norwegian's first European goal for almost three years.
United stormed forward in search of more goals. Boruc made fantastic saves to deny Saha and Wayne Rooney, but his best effort came when he kept out a header from one of his own players.
In the 83rd minute, Paul Telfer beat Saha to Darren Fletcher's cross and his header was brilliantly turned onto the post by the Celtic goalkeeper.