It was the fastest four-goal haul in the competition’s history and makes Lewandowski the top scorer in Europe this season with 10 goals in five games.
He also saw a first-half strike ruled out by VAR for a handball in the build-up.
Leon Goretzka and Corentin Tolisso also got on the scoresheet in Belgrade, but the night belonged to Lewandowski.
His spectacular goal burst in Belgrade was reminiscent of the five goals he scored in nine minutes in Bayern’s 5-1 Bundesliga win over Wolfsburg in September 2015.
The Poland striker now has 27 goals in 20 games for Bayern this season.
Having routed Tottenham 7-2 in London in October, when Serge Gnabry scored four goals, this was another memorable night for Bayern.
Bayern earned their fifth straight win in Europe, having already qualified for the last 16 in the previous round of matches.
They are on course to become the first German club to win all six group stage games before they host Spurs at the Allianz Arena in a fortnight’s time.
Interim coach Hansi Flick has now won all four matches – with 16 goals scored and none conceded – since Niko Kovac was sacked earlier this month.
Flick has been told he will stay in charge until at least the winter break, which starts Dec 22.
He opted to rest Joshua Kimmich and Gnabry, but Goretzka gave Bayern a dream start by heading home a curling Philippe Coutinho cross from the right on 15 minutes.
It could have been 2-0 to the visitors on 41 minutes when Lewandowski hit the top right-hand corner with a superb strike, but VAR spotted a Tolisso handball in the build-up.
Bayern should have been further ahead than 1-0 at the break having fired 15 shots at the Belgrade goal, while winning 12 corners with 71% possession.
Lewandowski’s stunning goal haul began when the referee awarded a penalty on 53 minutes for a handball by Red Star centreback Milos Degenek.
Lewandowski sent goalkeeper Milan Borjan the wrong way and then added his second seven minutes later with a classic striker’s goal – stabbing the ball home after a header from Tolisso.
Coutinho made way for Ivan Perisic for the final half hour.
Lewandowski wrapped up his hattrick inside 11 minutes by heading home a Benjamin Pavard cross, than bagged his fourth when he exchanged passes with Perisic and fired past the helpless Borjan on 68 minutes.
Having earlier been denied a goal when his shot clipped the bar, Tolisso finally netted when he capped a strong display with a goal in the 89th minute after good work by Perisic.