United were arriving ahead of their UEFA Champions League game at Bayer Leverkusen on Wednesday but their chartered plane had to abort its landing just 400m above the runway in Cologne.
The 185-seater Monarch Airbus 321 plane was unable to land because there was another plane on the runway and the pilot had to circle the airport before touching down 10 minutes late at 5.40pm.
Defender Rio Ferdinand wrote on Twitter afterwards: "Landed in Germany ... just. I've only just recovered after that choppy landing!"
And another member of the United camp told the Manchester Evening News: "It was quite frightening.
"We were coming in to land when we had to pull up suddenly, because it appears there was a plane on the runway.
"The captain came on the intercom to apologise for what happened and then we landed safely."
United manager David Moyes and other senior figures, including former chief executive David Gill, were on the flight with the squad, but Sir Bobby Charlton and former boss Sir Alex Ferguson were not on board, although they are both due to attend the match.
Eight Manchester United players were among the 23 people to tragically lose their lives in the Munich air disaster of 1958.
Charlton's fellow Munich survivor Bill Foulkes died on Monday at the age of 81.