Hansi Flick's side have now been victorious in each of their last seven matches in all competitions, while their third European victory of the season moved them into sixth position in the Champions League table on nine points.
The breakthrough came in the 13th minute, with a wonderful delivery from the in-form Raphinha being headed into the back of the net by Inigo Martinez, who had impressively freed himself at the far post.
Red Star Belgrade managed to answer back in the 27th minute of the contest, with Silas levelling the scores after bursting through on goal, with the attacker lifting the ball over the onrushing Inaki Pena.
Barcelona were back ahead in the latter stages of the first period, though, and it was another goal for Robert Lewandowski, with the striker converting from close range after a strike from Raphinha had struck the inside of the post.
The visitors then took the match away from Red Star Belgrade early in the second half, and the two assists came from Jules Kounde, who first set up Lewandowski to score from close range, before the Frenchman picked out Raphinha, and he continued his stunning start to the season with a brilliant finish into the corner to make it 4-1.
Kounde was at it again in the 76th minute, providing his third assist of the match, as the Frenchman's low cross was poked into the back of the net by substitute Fermin Lopez.
Red Star Belgrade managed to score a second in the 84th minute through Milson, but it was always going to be a case of too little, too late, as Barcelona comfortably triumphed in their fourth Champions League game of the season.
RED STAR BELGRADE VS. BARCELONA HIGHLIGHTS
Inigo Martinez goal vs. Red Star Belgrade (13th min, Red Star Belgrade 0-1 Barcelona)
Barcelona make the breakthrough in the 13th minute of the contest, as Martinez heads a delivery from Raphinha into the back of the net; Martinez broke free of the pack at the far post and directed his effort into the corner.
Silas goal vs. Barcelona (27th min, Red Star Belgrade 1-1 Barcelona)
Red Star Belgrade answer back in the 27th minute of the European affair, and the goal comes from Silas, who meets a pass from Rade Krunic before lifting the ball into the back of the net; impressive response from the hosts here!
43rd min: Red Star Belgrade 1-2 Barcelona (Robert Lewandowski)
Barcelona regain the lead in the latter stages of the first period, with Lewandowski turning into the back of the net from close range after a strike from Raphinha had come off the post and fallen kindly for the striker.
53rd min: Red Star Belgrade 1-3 Barcelona (Robert Lewandowski)
Lewandowski has his second of the night, and it is a simple finish for the striker, as he turns a cross from Kounde into the back of the net from close range; Red Star Belgrade now have a huge mountain to climb on home soil.
55th min: Red Star Belgrade 1-4 Barcelona (Raphinha)
Barcelona surely secure the points by scoring a fourth goal, with Raphinha excellently picking out the bottom corner from distance following another impressive cross from Kounde, who has come up with two assists tonight.
76th min: Red Star Belgrade 1-5 Barcelona (Fermin Lopez)
Kounde has a hat-trick of assists! Barcelona score their fifth of the match in the 76th minute, with Lopez carefully placing the ball into the corner after being found by his right-back; Barcelona are a serious team at the moment.
84th min: Red Star Belgrade 2-5 Barcelona (Milson)
Red Star Belgrade have their second of the night, with Milson curling one into the far corner of the net; it is a brilliant finish, but it will not earn the home side anything from this match.
MAN OF THE MATCH - JULES KOUNDE
Lewandowski came up with another two goals on Wednesday, but he just misses out on the man-of-the-match award due to the brilliant performance of his teammate Kounde.
The 25-year-old produced three assists against Red Star Belgrade, and his use of the ball was excellent all night, finishing with a pass success rate of 94%, with the Frenchman having one of his best games in a Barcelona shirt.
RED STAR BELGRADE VS. BARCELONA MATCH STATS
Possession: Red Star Belgrade 30%-70% Barcelona
Shots: Red Star Belgrade 4-21 Barcelona
Shots on target: Red Star Belgrade 2-9 Barcelona
Corners: Red Star Belgrade 1-7 Barcelona
Fouls: Red Star Belgrade 4-12 Barcelona
BEST STATS
WHAT NEXT?
Barcelona will now switch their attention back to La Liga, facing a trip to Real Sociedad on Sunday night; Flick's side are currently top of the La Liga table, nine points ahead of the reigning champions Real Madrid in second spot.
Red Star Belgrade, meanwhile, will continue their Super Liga campaign away to second-placed OFK Beograd on Sunday afternoon; Vladan Milojevic's side are the runaway leaders in Serbia's top flight, 10 points ahead of their opponents this weekend.