United star Cristiano Ronaldo was at the draw in Monaco, where he was named Uefa's European Club Footballer of the Year.
Ronaldo scored eight goals in the competition last season and also won the award for the forward of the year.
The other awards went to rivals Chelsea, with Petr Cech (goalkeeper), John Terry (defender) and Frank Lampard (midfielder) all voted the best at their position.
Chelsea, who lost last season's final to United, were the first side out of the hat and look to have been given the easiest task of progressing through to the knockout stages.
Among Chelsea's opponents will be French side Bordeaux, who are managed by former Manchester United defender and World Cup winner with France, Laurent Blanc.
Blues secretary David Barnard said: "It's a mixed bag but any team that reaches the group stages has to be given respect.
"We are totally focused on Champions League, the determination is there and they are completely focused on making the final again this year."
Manchester United will begin the defence of their trophy next month
Both Liverpool and Arsenal, who came through qualifying to make the group stages, face difficult opposition if they are to progress to the last 16 of the competition.
The draw means striker Fernando Torres will play against his former club Atletico for the first time since his move to Anfield in the summer of 2007.
Liverpool chief executive Rick Parry said: "It's not easy but the three teams are fairly close by and there's no challenging trips.
"There are no easy games - you just take it as it comes. We are just delighted to be in it."
Arsenal will have to travel to Turkey and Ukraine and director Danny Fiszman said: "One of the big issues for Arsene is the distances we have to travel. They are long trips and that's the most disappointing aspect of the draw.
"But we're optimistic and confident we will get to the knockout stages."
All five British clubs will be pleased to have avoided the likes of Bayern Munich and Juventus, who were potential opponents in the second pot of teams.
The first set of matches will be played on 16/17 September.
Full Champions League draw:
Zenit St Petersburg
Games will be played on the following dates:
30 September/1 October
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