On Wednesday, the German record-holding champions Bayern assumed control from the starting whistle while the visitors kept all their men behind the ball to make the hosts' life difficult, reports Xinhua news agency.
Bayern had to wait until the 21st minute before Philippe Coutinho produced the first clear-cut opportunity after coming close from the edge of the box.
Niko Kovac's men eventually broke the deadlock with 34 minutes played when Ivan Perisic's cross met Kingsley Coman, who marked the opener with a diving header from five meters.
After the restart, Bayern immediately took the reins and came to a number of chances but yet they were unable to overcome Belgrade goalkeeper Milan Borjan in the opening period of the second half.
Red Star Belgrade almost punished the host's wastefulness, but Marko Marin pulled wide from promising position moments later.
The visitors' chance was a wakening call for Bayern as Lewandowski capitalised on a moment of madness by the visitors' defence to scramble home the 2-0 lead from very close range.
However, the hosts were not done with the scoring and added another goal to their lead as substitute Thomas Muller benefitted on a chipped free kick to volley the ball past Borjan for the 3-0 final score.
With the win, Bayern Munich lead Group B with 3 points, followed by Olympiacos, Tottenham Hotspur and Red Star Belgrade.
In the other match, Bayer Leverkusen staged an uninspired performance and suffered a 1-2 loss to Lokomotiv Moscow.
With the result, Lokomotiv Moscow sits atop the standings with three points, followed by Atletico Madrid, Juventus and Leverkusen.
( With inputs from IANS )