With qualification behind Group A winners Paris St Germain already guaranteed, and perhaps having an eye on the upcoming El Clasico, Real Madrid coach Zinedine Zidane had freshened up the team.
Fellow Brazilian Vinicius, 19, staked his claim for more regular action with an instinctive finish after 64 minutes and Luka Modric clipped in a third during stoppage time.
Despite the defeat, Brugge will go into the Europa League knockout stages, having finished third in the group ahead of Galatasaray, who were thrashed 5-0 at PSG.
Militao headed an early corner wide before Brugge – seeking to secure Europa League qualification – went close when Percy Tau's close-range effort was saved by stand-in keeper Alphonse Areola.
The visitors created an opening after 15 minutes when Vinicius played in Jovic and the Serbian – who has scored just once following a big-money summer move from Eintracht Frankfurt – saw his goal-bound effort tipped over by Simon Mignolet.
Real Madrid slowly gained momentum as the half-hour mark passed, with Isco flashing a low drive just wide.
Brugge thought they had broken the deadlock just before half-time when Emmanuel Dennis flicked the ball in at the near post. The goal, though, was ruled out for offside in the build-up – with the decision upheld following a VAR review.
The Belgians made a fast start to the second half, with Hans Vanaken's volley blocked and Tau unable to force home the rebound.
Real Madrid took the lead in the 53rd minute when Modric released Odriozola on the overlap down the right and his cut-back was guided home from 15 yards by a clean strike from Rodrygo.
Brugge, though, were back on level terms within two minutes.
Dennis again proved too strong as he drove on into the left side of the Real Madrid penalty area before picking out Vanaken, who bent a curling shot into the far corner.
After another strong spell from the home side, Real Madrid got themselves back in front on 64 minutes.
Casemiro clipped a pass across for Jovic to head down to Rodyrgo in the six-yard box. The Brazilian, though, could not get hold of the ball, which bounced loose for Vinicius to slot into the corner.
With PSG comfortably ahead against Galatasaray in the other Group A match, Brugge were at least secure of further European progress, whatever the result at the Jan Breydel Stadion.
In stoppage time, a vintage finish from Modric, curling the ball past the goalkeeper from the edge of the penalty area, sealed a comfortable win.