Swansea moved out of the relegation zone as they comfortably dispatched a dismal West Ham side whose performance was so "embarrassing" manager David Moyes said it was "impossible to assess".
Ki Sung-yueng shot low into the corner from Andre Ayew's pass to open the scoring inside 10 minutes, before Mike van der Hoorn nodded in the South Korean's corner amid slack marking.
Between those two first-half goals, Winston Reid was put in a neck brace and taken off on a stretcher after suffering a bad injury in a goalmouth scramble.
He received treatment for 10 minutes after appearing to be knocked unconscious, and was taken to hospital for a scan.
After the break, West Ham crumbled further. Andy King tapped in the rebound from an Adrian save that denied Andre Ayew's header from another corner, before Cheikhou Kouyate tripped Ayew in the box.
Andre's brother Jordan Ayew coolly slotted the spot-kick into the corner to round off a vital win that was only slightly dampened by Michail Antonio's late consolation, the substitute winger finishing from Aaron Cresswell's delivery.
"The fans travelled through the snow to get here and we let them down badly," Moyes added.
"We had no intensity in our play, we thought we could just stroll about."
Meanwhile, Carlos Carvalhal's impressive work continues. His team are now up to 13th in the table - climbing above the Hammers on goal difference - having been bottom when he took over in late December. This was a seventh consecutive home win in all competitions.
West Ham drop to 14th, with both sides still only three points clear of the bottom three.
The anatomy of a goal
Just after the break, with David Moyes' thoughts on being two goals down at half-time presumably still fresh in their minds, West Ham shipped a third.
Andre Ayew rose to head from another Ki corner, and Adrian reacted well to keep it out but palmed his stop straight into Javier Hernandez's face. Out the ball came - dropping to the one white shirt in a position to tap home, as the markers around him slumped.
King's goal was his first since joining on loan from Leicester, and it summed up West Ham's performance in one action. They were sluggish, unfortunate, beaten.
In the first half, Adrian had been slow to get down and reach the Ki shot from range that crept steadily into the corner after nobody closed him. Then came the injury to Reid that left the Hammers with a makeshift defence, four regular full-backs across the line.
That might explain something of the poor marking from which Van der Hoorn profited for the second goal that sent Swansea clear and never looking back, but not once did it feel like West Ham were capable of finding a response to adversity.
With just nine games remaining before the end of the season - and just two of those coming against teams also in danger of relegation (Stoke and Southampton), this is what will concern Moyes most of all.
Moving on up - and staying there
Swansea reached the FA Cup quarter-finals for the first time since 1964 with victory over Sheffield Wednesday in a dire match on Tuesday evening. Afterwards, Carvalhal said staying in the Premier League was still the club's priority.
Perhaps that's why this game was so much more entertaining. The home side bristled with confidence and adventure as they took full advantage of their opponents' apparent surrender.
These were three more points won in a remarkable run under manager Carvalhal - and despite last weekend's bruising 4-1 defeat at Brighton.
The Portuguese has transformed his side since taking over from Paul Clement, winning 17 points from nine league games in charge. Clement had won 12 points from the season's first 18.
On average, the number of points to guarantee survival in the previous 22 top flight seasons is 36.6. On this evidence, with matches against Huddersfield and Southampton to come next, they will surely easily surpass that total.
Deadly from the spot - the stats
* Swansea have scored 20 of their 21 Premier League penalties. Of teams to have taken at least five, that is the best success rate in the competition's history (95%).
* Ki Sung-Yeung has had a hand in three goals in his past four PL appearances for Swansea (two goals, one assist), having previously gone 14 games without registering a single goal or assist.
* West Ham have conceded 37 goals away from home in the league so far this season, more than any other side in the competition.
* Swansea have scored 14 goals in nine league matches under Carvalhal, having scored just 11 in their previous 20 this season before his appointment.
What the managers said
Swansea boss Carlos Carvalhal: "I am delighted, I am proud of my players.
"There is no secret to success. It is hard work, organisation, patience, dedication and correcting mistakes. We are better and better, we are really strong at the moment.
"I put the players in the centre of everything. If you don't have players who follow your ideas you can't achieve anything.
"We have trained hard at corners this week, movement, and we scored twice from them. Now we must create another strategy for the next game."
West Ham manager David Moyes: "We conceded from two corners and a penalty. We had no intensity in our play, we thought we could just stroll about.
"Winston Reid is conscious, he was knocked out, we are more worried about the injury to his knee than the knock to his head. He's an important player for us so we hope it's not too bad.
"We should have saved the first one, we should head the second one away and the third one is a corner and the fourth is a penalty.
"We have to be much, much better. We have the players who can do it but getting them to perform is my job."
What's next?
Swansea play away to Huddersfield in the Premier League next Saturday. West Ham host Burnley in their next game on, also on Saturday.
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