The La Liga leaders stuttered slightly in the first half in the Serbian capital, but soon took complete control and showed little mercy on the side that now sit second-from-bottom in the 36-team standings.
Robert Lewandowski put Barca back in front after Inigo Martinez's opener was cancelled out by Silas, and the Polish striker netted again shortly after half time to take his Champions League career tally to 99.
Raphinha and Fermin Lopez added further goals before a late Red Star consolation, but the commanding victory moved them into the top eight and, should they hold onto that spot, instant qualification for the last 16.
Flick's Barca set impressive club record
The five goals scored in Belgrade mean that Barcelona have netted 55 in their opening 16 games across all competitions this season - a new all-time record.
Tonight's tally means that the current crop have beaten the previous best, which was held by the 1950-51 squad, who managed 54 across the same number of matches to begin the campaign.
Hansi Flick's instant impact at the club has been transformative on the pitch, and this record illustrates the influence he is having on a squad that lagged a long way behind rivals Real Madrid last season.
Barca finished 10 points adrift of Los Blancos, who cruised to the La Liga title last season, but it is the Catalans who currently hold top spot after 12 games of the current campaign.
Real have a game in hand, but the 4-0 derby demolition in El Clasico two weeks ago means Flick's side have a nine-point lead at the La Liga summit.
Notable moments from Barca's season so far
Of all 55 goals, the four most cherished will be those netted at the Bernabeu against the eternal enemy, but there have been plenty of other eye-catching performances from Barcelona this season.
No team has scored more than Barca's 15 in the Champions League, and their 4-1 thrashing of Bayern Munich on matchday three will surely go down as one of the performances of the league phase, as Raphinha's hat trick put Vincent Kompany's men to the sword.
The win over Red Star was their fourth five-star showing this season, as Sevilla, Villarreal and Young Boys have also shipped five against Flick's fearsome attack, while poor Real Valladolid were thrashed 7-0 at Montjuic back in late August.
As well as the El Clasico success, derby delights have been commonplace too, as Espanyol were beaten 3-1 at the weekend, and Barca avenged back-to-back defeats to Girona last season by beating their Catalan neighbours 4-1 earlier in the campaign.