Michael Laudrup's side impressed at the Emirates with fine passing and confident play, with his Spanish striker scoring two excellent goals to compound Arsene Wenger's misery
The Spanish forward found himself through on goal with three minutes remaining and beat Wojciech Szczesny with a fine curling finish before sealing the points in injury time with another confident strike after a slip from Carl Jenkinson gave him sight of the target again.
In-form striker Olivier Giroud was replaced by Gervinho having played seven consecutive matches for the Gunners, while Laurent Koscielny and Bacary Sagna missed out at the back.
Michael Laudrup's team went to the Emirates after a fine 3-1 win over high-flying West Brom, and started the game the stronger of the sides despite the absence of Wayne Routledge and Pablo Hernandez.
Thomas Vermaelen made a fine tackle in the fourth minute to deny Itay Shechter with the striker through on goal, while the home side had to scramble the ball away a minute later in a frenetic opening.
Laudrup's side could have taken the lead on 13 minutes - Shechter found Angel Rangel free on the right of the penalty area with a cute pass, but Szczesny brilliantly denied the Spaniard twice.
Nathan Dyer had a glorious chance on the half hour after the forward burst on to Michu's knock-down, but Vermaelen did superbly to deny the winger by blocking his goal-bound effort from 16 yards.
Gervinho had a chance to head home from 12 yards but could only glance his header away from goal as the home side departed to boos at the whistle.
However, Wenger must have had strong words for his players at half time as the Gunners came out after the interval full of vigour, with Cazorla's low 18-yard drive stinging the hands of Gerhard Tremmel on 47 minutes.
Wenger's much-needed changes came on 65 minutes, with Lukas Podolski making way for Giroud, while the disappointing Gervinho left the pitch for Alex Oxlade-Chamberlain to inject some pace into the side.
Controversy ensued on 69 minutes as Chico Flores got the slightest of touches on the ball when Giroud looked to be through on goal, with referee Mark Clattenburg doing well to spot the challenge as Wenger fumed on the touchline.
Two minutes later Cazorla showed fine technique to control a left-footed volley from 18 yards, but Tremmel did well to hold onto the swerving effort.
Great interplay from Luke Moore and Dwight Tiendalle saw the defender through on goal on the right, with Szczesny again on hand to produce a fine stop for the hosts.
Vermaelen powered a header at goal from 12 yards out as the hosts went searching for a valuable winner, but Michu left Arsenal's title hopes in tatters minutes later after striking to secure the visitors three points.
Michu would double the Swans' advantage before the end and the home fans' exasperated reaction at the full-time whistle suggests the transfer window can't come soon enough for the Gunners.