The former Wolves man made a splash on his Premier League debut for the Black Cats but the home side pegged them back and held on, despite Chico's reckless sending-off
After a dull opening half an hour, Steven Fletcher latched on to Ashley Williams' slack back-pass to strike on his league debut but Wayne Routledge's thumping effort equalized just six minutes later before the Scot bundled home a second just before the half-time whistle.
After intense second-half pressure the home side equalized through Michu and managed to hold on to a point, despite Chico's sending off.
Michael Laudrup's side were unchanged from their 3-0 win over West Ham last time out in the league while the Black Cats gave Premier League debuts to Steven and Adam Johnson.
The Swans dominated the early going without threatening, but were dealt a blow after 20 minutes as Neil Taylor was forced off with a nasty-looking ankle injury picked up in a challenge with Craig Gardner, and was replaced by youngster Ben Davies.
Both sides struggled to find a breakthrough and Sunderland were forced into a change of their own after 38 minutes as Lee Cattermole was replaced by David Meyler after picking up a dead leg in a clash with Routledge.
With the game meandering towards half-time the away side struck out of nowhere, Williams attempted a first-time back-pass to Vorm but only scuffed his effort into the path of Fletcher who raced towards goal and finished coolly past the Dutchman.
The game suddenly sprung into life and after James McClean volleyed over the Swans found themselves level. After a patient build-up on the edge of Sunderland's area, Nathan Dyer sprung into life, chipping a delightful ball into the path of Routledge who smashed past Simon Mignolet.
Six minutes of added time had been added on due to the game's injuries and it was enough time for the Black Cats to regain the lead. Chico looked to clear Sebastian Larsson's deep free kick to the back post but he was charged down by Fletcher and the ball was bundled in by a combination of the two players.
The Swans came out for the second half on the attack, Leon Britton hit an effort over from eight yards from a well-worked corner before Mignolet was called upon to keep out a fierce Dyer effort at his near post.
Laudrup's side continued pushing on and went close again on the hour. Mignolet could only palm Jonathan de Guzman's free kick into the path of Angel Rangel but the Belgian recovered well to keep the right-back's follow-up out and the ball was cleared.
Michu was then guilty of wasting a gilt-edged chance, the Spaniard heading into the ground and over from 10 yards when it looked easier to score, but the former Raye Vallecano man soon made up for his miss.
De Guzman was given too much room deep in midfield and swung a perfect ball onto the head of Michu who planted his headed effort past Mignolet for a richly deserved Swans equalizer.
Martin O'Neill responded by replacing the tiring Fletcher with Louis Saha and the Frenchman was immediately into the thick of the action as Chico made a wild, head-high lunge at the Frenchman, and was sent off, despite making minimal contact with the Frenchman.
Michael Laudrup brought on Alan Tate for Routledge in order to shore up the defense, with just under 15 minutes remaining in the match.
The Swans controlled the game out for much of the rest of the match and could have snatched a point as Michu scrambled an effort in the six-yard box but the Black Cats held out for a point.
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