Referee Anthony Taylor took centre stage with key decisions to award Liverpool a second-half penalty after earlier allowing West Ham's first-half equaliser to stand.
The winning goal came via the boot of captain Steven Gerrard from the spot in the 71st minute after Adrian was penalised for bringing down Jon Flanagan, with replays showing the West Ham goalkeeper touched the ball before the pair collided.
It was Gerrard's second penalty of the game, the veteran midfielder having broken the deadlock in a closely-fought first-half after James Tomkins handled as Luis Suarez turned him in the box.
But West Ham were back on level terms as Guy Demel poked home in first-half added time when Simon Mignolet failed to gather Mark Noble's corner, under huge pressure from Andy Carroll.
Referee Taylor gave the goal, only to see his assistant was flagging. Replays of the incident, which were also inadvertently shown on the big-screen in the ground and seen by the players, showed Carroll's flailing hand clearly connect with Mignolet's head, but after a conversation between the officials, the goal was allowed to stand.
Now Liverpool can prepare for an Anfield clash with trophy rivals Manchester City next Sunday knowing another win - that would be their tenth in succession - will bring a first league title for 24 years even closer.
Luis Suarez sent out a warning shot inside the opening three minutes as he bent a 30-yard free-kick just over the top of Adrian's crossbar.
He went closer still with his next effort as he clipped a shot over the West Ham keeper, only to see the ball come back into play off the goalframe.
All of Liverpool's threat was coming courtesy of 29-goal Suarez, who looked to whip the ball past Adrian from another opening, with James Tomkins bravely heading the shot behind.
With just a minute remaining of the first half Suarez was released by a sumptuous Gerrard pass and, as the Uruguay international looked to cut inside, the ball hit the arm of Tomkins, with referee Taylor pointing to the spot.
Gerrard stepped up confidently to send Adrian the wrong way and put the visitors ahead.
But West Ham did not let their heads drop and levelled in controversial circumstances on the stroke of half-time.
Noble's corner was aimed towards Carroll, whose hand caught Mignolet in the face as the goalkeeper looked to take possession, with Demel on hand to turn the ball in.
Taylor gave the goal after a long consultation with assistant referee Stuart Burt and was in deep discussion with Brendan Rodgers as the teams disappeared down the tunnel at half-time.
The Hammers started the second half under pressure but managed to soak up anything Liverpool could throw at them before the hour-mark, before creating the best chance to take the lead.
Demel and Diame combined down the right before the latter swung in a cross attacked from deep by Carroll, whose header cracked off the crossbar with Mignolet well beaten.
Daniel Sturridge burst past Matt Taylor and Winston Reid as the half went on but once again his direction was slightly askew and he failed to test Adrian.
But Liverpool were rewarded a second penalty as Adrian was adjudged to have brought down Flanagan, despite getting a hand to the ball before bundling into the full-back.
Gerrard again made no mistake from 12 yards, scoring a pair of penalties for the second time this season.
Sturridge finally managed to find the target with 10 minutes remaining as he fired in a dipping volley which was too close to Adrian to cause the Spaniard any real trouble.
Suarez once again hit the crossbar, this time with a swerving lob, and was denied by the feet of Adrian late on, with Sterling also forcing a smart save in injury time.