Aaron Cresswell scored his first goal for the club to make it 13 points from five home matches for Sam Allardyce's side, as Alan Pardew sees his own good run come to an end
In Saturday's meeting of two of the Premier League's form sides, Newcastle were searching for their seventh straight triumph in all competitions.
Yet they were undone by a West Ham team that has now picked up 13 points from their last five home games, with left-back Cresswell proving the matchwinner.
The 24-year-old - who joined West Ham from Championship side Ipswich Town in July - settled the encounter with a close-range strike in the 56th minute as Alan Pardew's defence was breached for the first time this month.
In a game of few clear-cut opportunities, Newcastle could not find the goalscoring touch and their frustration was compounded when midfielder Moussa Sissoko was sent off for picking up two yellow cards in two minutes.
West Ham's sixth league win of the campaign sees them leapfrog the Magpies in the table.
The visitors made a bright start to proceedings and went close inside the first three minutes as the in-form Ayoze Perez shot wide of the far post following good work down the right from Yoan Gouffran.
Downing drew a comfortable stop from Elliot with a low strike, but that was the best West Ham could offer in the opening stages despite Newcastle allowing their hosts to enjoy the majority of possession.
Perez was the most dangerous player on the pitch in a dour first half and provided another example of his attacking threat by cleverly flicking into the side-netting after a near-post cross from Sammy Ameobi, who then curled wastefully wide in the 44th minute.
Elliot's goal was rarely threatened by the hosts, but he was relieved to concede only a corner early in the second half when he charged from his line to try and clear a relatively aimless free-kick, only to miss the ball completely.
Newcastle's keeper could then do nothing to prevent Cresswell from putting West Ham ahead.
The full-back capitalised on some sloppy Newcastle defending, latching on to a mis-hit shot from Cheikhou Kouyate and slotting into the bottom-left corner from close range.
Pardew threw on Papiss Cisse and Remy Cabella in response to falling behind, yet the changes had little impact for a Newcastle side that laboured in search of an equaliser.
And a disappointing afternoon for Pardew's men worsened courtesy of two moments of madness from Sissoko, the France international seeing red for a late challenge on former Newcastle striker Andy Carroll less than 30 seconds after being cautioned for kicking the ball away.
Sissoko's early exit effectively ended any hope of a Newcastle fightback and leaves an already injury-hit side facing further problems ahead of Tuesday's visit to Burnley.