Premier League: Hull 0 Man City 2
Captain Kompany saw red in the opening 10 minutes on Saturday after hauling down Nikica Jelavic as the striker looked set to race through on goal, but David Silva's superb curling effort handed the visitors the lead just four minutes later.
Pablo Zabaleta hit the crossbar before the break as Manuel Pellegrini's reorganised charges continued to underline their attacking prowess, but the hosts also had their moments.
Jelavic, restored to the starting line-up along with Shane Long after Hull's FA Cup exploits, was frequently a threat and David Meyler had an effort ruled out for offside.
Hull upped the pressure after the interval but were undone by Edin Dzeko's late goal, assisted by Silva, as their search for a home league win in 2014 continues.
Pellegrini may have seen his dreams of a historic quadruple go up in smoke this month, with his side's FA Cup and UEFA Champions League campaigns brought to an end in the space of four days, but the top-flight crown remains within his reach.
Steve Bruce's side remain uncomfortably close to the bottom three but made a confident start, Jelavic seeing his pull-back cleared away by Gael Clichy before the Croatia international was denied by an offside flag as he raced clear of the visiting defence.
And the game appeared to swing in the hosts' favour when Kompany resorted to bringing Jelavic down following a physical tussle 30 yards from goal.
Referee Lee Mason opted to send the Belgian off despite his protestations that he had been fouled first, with Kompany kicking out at a wall as he stormed down the tunnel.
But Silva turned Manchester City's frustration into joy when he seized on the opportunity to shoot from 25 yards and produced a glorious effort that swerved inside Allan McGregor's right-hand post, ending his 17-game goal drought.
The goal buoyed Pellegrini's men but they still had defending to do when Long's searching right-wing cross almost found Jelavic, with Martin Demichelis doing just enough to snuff out the danger.
Zabaleta struck the underside of the crossbar with a dipping volley at the other end on the half-hour mark, the ball bouncing on the line and out to safety.
Meyler saw an effort ruled out for an offside in the build-up and Jake Livermore flashed a low, skidding effort narrowly wide of Joe Hart's upright on the stroke of half-time as Hull sent a timely reminder that they were still very much in the contest.
Jelavic was again involved early in the second half, making poor contact with half-time substitute Sone Aluko's dangerous free-kick to leave Hart with a simple save.
George Boyd, on for Meyler just before the hour, took a tumble in the box with Hart in close attention but Mason awarded a goal-kick and booked the goalkeeper for his part in the confrontation that followed.
Dzeko put the game to bed in stoppage time after wasting an earlier effort as the visitors went some way to avenge defeat on their last trip to the KC Stadium and put the bitter disappointment of Champions League elimination behind them.