Arsenal goalkeeper Cech's miscued attempt to clear allowed Jordan Ayew to slot home Swansea's second goal at the Liberty Stadium.
It was the killer blow for Arsenal, who had already seen Sam Clucas equalise just one minute after Nacho Monreal put the visitors ahead from Mesut Ozil's pass in the 33rd minute.
Clucas claimed his second four minutes from time to finish off Arsenal, who have now won just once in their last five league games.
The result leaves Gunners manager Arsene Wenger still searching for his first away win of 2018, despite a second- half debut for substitute Henrikh Mkhitaryan following his move from Manchester United.
The arrival of Borussia Dortmund striker Pierre-Emerick Aubameyang, set to sign on Wednesday's transfer deadline day, can't come soon enough for Wenger.
The result also completes remarkable back-to-back victories for Swansea over Liverpool and now Arsenal.
They are out of the relegation zone and have transformed their season with three wins in five league games under new boss Carlos Carvalhal.
At the John Smith's Stadium, Liverpool got back on track in emphatic fashion thanks to goals from Emre Can, Roberto Firmino and Mohamed Salah in a 3-0 win.
But Jurgen Klopp's men dominated possession and from the moment Emre Can rocketed home a long-range opener in the 26th minute the result was never in doubt.
Roberto Firmino extended Liverpool's lead on the stroke of half-time when he somehow squeezed the ball home from the goal-line after being played in by a clever Sadio Mane back- heel.
Klopp had left Dutch defender Virgil van Dijk on the bench, but the absence of the club-record signing was hardly noticed as the Reds continued in cruise control, with Salah completing the scoring from the penalty spot in the 78th minute after Can was tripped up in the box.
Roy Hodgson's much-improved Eagles grabbed the lead in the 24th minute when Christian Benteke rose to steer an Andros Townsend cross home from close range.
But the Hammers were level just before the break after Javier Hernandez was brought down by James Tomkins in the box.
Hernandez came closest to a winner when his 51st-minute effort was brilliantly saved by Wayne Hennessey.
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