It was Brazilian Pato who was the show's central figure as he produced two clinical finishes and a dubious fall to turn the most eagerly-anticipated Milan derby in years the way of the rossoneri.
Substitute Antonio Cassano added a late penalty after he tumbled under a Javier Zanetti challenge.
The result leaves Milan five points clear of the champions at the top of Serie A with only Napoli capable of closing that gap if they beat Lazio on Sunday.
However, what had until then been a pulsating affair was ruined somewhat on 54 minutes with the controversial dismissal of Chivu.
Milan were off to a flying start as Pato put them in front in the first minute.
Gennaro Gattuso played Robinho in behind a static Inter backline but although Julio Cesar came out bravely to block, Pato was following up to slot home the loose ball.
The action was fast and furious early on with Inter pair Goran Pandev and Wesley Sneijder failing to trouble Christian Abbiati with shots from distance.
A point-blank volley from Clarence Seedorf then appeared to hit Maicon's hand in the box but the referee waved away appeals for a penalty.
Inter had a great chance to level on 19 minutes as Pandev played in Giampaolo Pazzini, who got past his marker courtesy of a lucky bobble, only to blast his shot straight at Abbiati.
Pato had a shot deflected just past the post and then got past Chivu only for Andrea Ranocchia to come across quickly to snuff out the danger.
On 38 minutes a Mark Van Bommel blast from distance took a wicked deflection off Chivu and beat Cesar but crashed back off the bar and into the goalkeeper's relieved arms.
A minute later came the second most controversial episode of the match as Thiago Motta headed Sneijder's corner goalwards only for Abbiati to claw the ball away from on the line.
TV replays were inconclusive but at least three-quarters of the ball was over the line, if not more.
Late in the half Samuel Eto'o missed an absolute sitter from three yards but was anyway given offside.
But the game hinged on a scandalous decision by the referee nine minutes into the second period as Pato was sent clear of the backline and ran across Chivu before theatrically flopping to the ground.
The referee adjudged the Romanian to have brought him down despite barely a hint of contact -- and that only provoked by the direction of Pato's run -- and showed him a straight red.
Cesar stood up strong to Thiago Silva's free-kick and Inter survived, but only temporarily.
Pato stayed just onside on 62 minutes with Inter's backline stretched to head home Ignazio Abate's cross from five yards.
Milan threatened to run riot and Cesar had to make a crucial block to deny Robinho before Ranocchia deflected a Kevin-Prince Boateng effort behind.
Substitute Ivan Cordoba then got back to deflect Robinho's shot behind after the Brazilian over-complicated matters in dummying Cesar rather than shooting sooner.
Cesar then saved again from Robinho as he once more strayed beyond Inter's defence.
But the goalkeeper could do nothing about Cassano's penalty a minute from time after he too fell like a dying swan under Zanetti's challenge.
However, he then managed to get himself sent off as he was booked twice within a minute, first for removing his shirt while celebrating and then for tripping Cordoba.