On top of that joy for the hosts, the victory on Sunday increased Madrid's league lead over Barcelona to five points after 10 rounds.
There was a scuffle between the bench players from both teams after Barcelona's Pedri was sent off just before the final whistle.
The altercation continued after the game ended, with Madrid forward Vinicius Junior and Barcelona star forward Lamine Yamal — who was successfully contained by Madrid defenders during the match — among those involved.
Vinicius and Yamal — whose pre-match comments had upset some Madrid fans — exchanged words a couple of times during the game.
Vinicius gestured against Yamal, using his hands to hint that the Barcelona star was talking too much.
Yamal and Madrid goalkeeper Thibaut Courtois also got into an altercation after the match.
Vinicius was visibly upset when he was replaced by Rodrygo in the 72nd minute, going straight to the locker room before rejoining his teammates on the bench a few minutes later.
France forward Mbappe, who failed to convert a second-half penalty, had put Madrid ahead halfway through the first half.
That came after the VAR had reversed a penalty-kick call and a goal for the hosts in the opening 15 minutes at the Santiago Bernabeu Stadium.
Mbappe opened the scoring in the 22nd minute after an assist by Bellingham, but Barcelona equalised through Fermín Lopez in the 38th and Bellingham netted the winner in the 43rd.
Mbappe had his penalty kick saved by Barcelona goalkeeper Wojciech Szczesny in the 52nd.
Madrid have now won 12 of their 13 matches this season, with the lone setback a 5-2 loss at Atletico Madrid in the Spanish league in September.
But it was the third loss of the season for Barcelona in all competitions. They had also lost this month 2-1 to Paris Saint-Germain at home in the Champions League and 4-1 at Sevilla in the Spanish league.
Mbappe increased his league-leading tally to 11 goals in a breakaway, calmly finding the net with only Szczesny to beat after a nice through ball by Bellingham. It was his 16th goal in his last 13 matches with club and country.
Barcelona did equalise after a mistake by Madrid midfielder Arda Guler, who lost possession near the area to allow Marcus Rashford to make a run through the left side and find Lopez for an easy strike from close range.
Bellingham put Madrid ahead again five minutes later, finding the open net from near the goal line after Madrid defender Eder Militao headed the ball towards the goal following another nice run and cross by Vinicius.
