The Hammers had the better chances of the early stages, but they took the lead in contentious circumstances as Mark Noble calmly converted from the penalty spot past Steve Harper after 26 minutes.
Hull's second-choice goalkeeper had replaced Alex Bruce after Allan McGregor saw red for bringing down Mohamed Diame inside the area, although replays suggested that the midfielder had handled the ball in the build-up.
Despite their numerical deficit, the visitors were level almost immediately after the break as Tom Huddlestone drove a free-kick off the unassuming Nikica Jelavic and Adrian was left helpless as the ball rolled past him.
However, Sam Allardyce's side found themselves back in front just six minutes later, with Chester making a hash of his attempted clearance, slicing Guy Demel's inswinging cross over the head of Harper.
The majority of home fans inside of Upton Park were restless throughout the night as their side toiled against a side reduced in numbers for over an hour, with a small number jeering the players as they left the pitch at both half time and at the end of the game.
After an uneventful opening spell, with only a low shot from Hammers' winger Stewart Downing testing either goalkeeper, the game burst into life on 20 minutes.
West Ham, who had seen James Collins forced off with injury early on, were awarded a penalty as McGregor bundled into Diame after good work from Noble had freed his midfield partner in the box.
The ball hit Diame on the arm before McGregor made contact but referee Mike Dean pointed to the spot after the Senegal international 's goal-bound effort was hacked off the line by Maynor Figueroa.
McGregor required treatment following the collision but was shown a red card once he had recovered, becoming the first goalkeeper in almost a decade to be sent off twice in a Premier League season.
Bruce replaced son Alex with substitute goalkeeper Harper, who could do nothing to prevent Noble putting the home side ahead from 12 yards.
Hull had a good chance to draw level on a rare foray forward as Ahmed Elmohamady broke down the right flank before crossing to Huddlestone, who could not keep his volleyed effort down.
Andy Carroll saw a header bounce up and over the bar as West Ham continued to struggle to make their numerical advantage tell.
The Tigers were enjoying some success with Chester heading a Huddlestone free-kick well over the bar in stoppage time before Figueroa finally drew a save out of West Ham goalkeeper Adrian with a fierce long-range effort.
The depleted visitors started well after the interval and drew level in fortuitous circumstances as Huddlestone's driven free-kick deflected off Jelavic and flew past the wrong-footed Adrian.
If Hull's goal had a hint of luck, they had used their quota for the evening as Demel's cross hit Chester's knee and looped over his own goalkeeper to put West Ham back in front just six minutes after Jelavic's leveller.
The own goal did not give Allardyce's side the impetus to go on and put the game to bed, though, with Downing firing high and wide when well-placed.
Kevin Nolan forced Harper into a sprawling stop with a little under 10 minutes remaining as the resistance offered by the 10 men of Hull started to wane.
Huddlestone was once again afforded enough space to arrow a shot towards goal, with Adrian opting for the safety-first approach of pushing the ball away.
The search for a deserved equaliser ultimately proved fruitless for the Tigers, who slipped to their first away league defeat in almost two months.
The win takes West Ham above Hull and up to 11th in the table with nine points now separating them from 18th-placed Sunderland, who they face at the Stadium of Light on Monday.