The British pair will also face Spaniards Villarreal and Danish side Aalborg in Group E.
Chelsea were drawn with Roma, Bordeaux and Romanian minnows CFR Cluj, while Liverpool will meet PSV, Marseille and Atletico Madrid in a tough group.
Arsenal will play Porto, Fenerbahce and Dynamo Kiev in Group G.
The four English sides were guaranteed to be kept apart after being made top seeds, but Celtic could have faced any one of them after being drawn in the third pot of clubs. They will get their chance to test themselves against the English champions in two mouth-watering clashes.
The fixtures will pit Celtic manager Gordon Strachan against his one-time boss Sir Alex Ferguson. The pair met in the Champions League group stages in 2006/07, with Celtic winning 1-0 at home but Manchester United edging the fixture 3-2 at Old Trafford.
"It was a great event the last time we played," said United chief executive David Gill. "Both clubs have passionate fans and we're looking forward to it.
Last time they won (at Celtic) to qualify and we went down to the last game to ensure we went through.
Group A
CHELSEA
Roma
Bordeaux
CFR Cluj
Group D
LIVERPOOL
PSV Eindhoven
Marseille
Atletico Madrid
Group E
MAN UTD
Villarreal
CELTIC
Aalborg
Group G
ARSENAL
Porto
Fenerbahce
Dynamo Kiev