UEFA confirmed that a "clerical error" occurred where United were mistakenly omitted as potential opponents for Atletico in the original draw.
UEFA director Giorgio Marchetti said ahead of the redraw: "I apologise we have had to redraw to do the round-of-16 draw. There were some errors made in the original draw."
Cristiano Ronaldo and Lionel Messi had appeared set for another Champions League showdown during Monday's initial draw, when United were eventually paired with PSG, having earlier erroneously been placed as opponents for Villarreal, whom they could not have met, having been in the same group as Unai Emery's side.
Group winners will be away in the first leg, with ties to be played on Feb. 15-16 and Feb. 22-23. Return legs will be played on March 8-9 and March 15-16.
The away goals rule has also been abolished in European tournament knockout ties this season.
The rule, which had been in place since 1965, was used to determine a winner when teams were level on aggregate after a two-legged match. The team that had scored more goals as the away side were awarded the victory. But from now, this will be replaced by extra time and penalties.
UEFA will draw the quarterfinal and semifinal ties on the same day for this season, scheduled for March 18.
The final will take place on Saturday, May 28, at Krestovsky Stadium in Saint Petersburg, Russia.
FC Salzburg vs.Bayern Munich
Sporting CP vs. Manchester City
Benfica vs. Ajax
Chelsea vs. Lille
Atletico Madrid vs. Manchester United
Villarreal vs. Juventus
Inter vs. Liverpool
Paris Saint-Germain vs. Real Madrid
Initial round-of-16 draw, declared void
Benfica vs. Real Madrid
Villarreal vs. Manchester City
Atletico Madrid vs. Bayern Munich
FC Salzburg vs. Liverpool
Inter Milan vs. Ajax
Sporting CP vs. Juventus
Chelsea vs. Lille
Paris Saint-Germain vs. Manchester United