A minor injury to Thibaut Courtois meant that Petr Cech made his first league start of the season in goal for Chelsea, who were bidding to bounce back from last weekend's 2-1 defeat at Newcastle.
Jose Mourinho's side went ahead in the seventh minute at Stamford Bridge, with Hazard ghosting in from the right flank to head home Oscar's in-swinging cross from the left.
Hull lost Michael Dawson to injury shortly after, with Alex Bruce coming on, and saw Huddlestone given his marching orders on the hour for a sly stamp on Filipe Luis.
Costa made it 2-0 eight minutes later, gathering a pass from the influential Hazard and drilling a shot past Allan McGregor to claim his 12th goal of the campaign.
Manchester City registered a seventh successive win in all competitions, but victory at the King Power Stadium came at the cost of injuries to striker Dzeko and captain Kompany.
Having previously lost top scorer Sergio Aguero to a knee injury that will keep him out until next year, City saw Dzeko withdraw from the starting XI after hurting himself in the warm-up.
It meant that 18-year-old Spaniard Jose Pozo started up front on his own, but after a laboured start, the champions went ahead in the 40th minute when Frank Lampard tucked away a pass from Samir Nasri.
It was the former Chelsea midfielder's 175th league goal, taking him level with Arsenal great Thierry Henry in fourth place in the all-time Premier League scoring chart.
But the afternoon ended on a sour note for Manuel Pellegrini's side when Kompany, who was making his return from a hamstring injury, had to go off after apparently aggravating the problem in the latter stages.
West Ham remain fourth after their draw at Sunderland. Stewart Downing equalised for the visitors with a low, deflected shot shortly after Jordi Gomez had put the hosts ahead from the penalty spot in the 22nd minute.
Burnley climbed out of the relegation zone by beating Southampton 1-0 at Turf Moor.
Southampton winger Dusan Tadic saw a penalty saved by Tom Heaton, enabling Ashley Barnes to claim the winning goal with a deflected shot 17 minutes from time.
Craig Gardner scored the only goal as West Bromwich Albion won 1-0 at home to Aston Villa, who had West Brom old boy Kieran Richardson dismissed for a studs-up lunge on Stephane Sessegnon.
Meanwhile, Stoke City striker Peter Crouch swiftly cancelled out a headed 11th-minute opener by James McArthur as Mark Hughes's team claimed a 1-1 draw at Crystal Palace.
-AAP