Gomez fired home his first Premier League goal for Sunderland with a clinical left-foot finish 11 minutes from time to put the Wearside club back in front and Fletcher's second of the night on Monday sealed the points.
And nobody would have been more relieved than team-mate Brown, who had drawn Palace level with an own goal in the second half after Fletcher had given Gus Poyet's side a first-half lead.
Brown was dropped to the substitutes' bench after his poor backpass gifted Arsenal an opening goal in their 2-0 win at the Stadium of Light last weekend, having also featured in an 8-0 thumping at Southampton the previous weekend.
The former Manchester United defender came on as a first-half substitute at Selhurst Park following an injury to Patrick van Aanholt and had another moment to forget as he found the back of his own net.
Gomez and Fletcher struck in the last 11 minutes, though, and Mile Jedinak was sent off for a second bookable offence to add to Palace's misery, the home side having also had two penalty appeals turned down. Sunderland move up to 15th spot as a result, with Leicester City dropping into the bottom three below Palace on goal difference.
Poyet dropped goalkeeper Vito Mannone following his howler against Arsenal last weekend on the back of conceding eight in the thrashing at Southampton, with Costel Pantilimon making his first Premier League start for Sunderland. Recent signing Anthony Reveillere was also handed his debut for the Wearside club.
Scott Dann returned to the Palace starting line-up after recovering from a knee injury, with Adrian Mariappa dropping to the bench in the only change to the side that drew at West Brom.
Neil Warnock felt Palace were on the wrong end of poor decisions from officials in their last two games and he was entitled to feel aggrieved again when referee Phil Dowd failed to award a penalty inside 30 seconds.
Santiago Vergini appeared to upend Frazier Campbell after the former Sunderland striker outpaced him to race onto a Marouane Chamakh flick on, but Dowd waved play on.
The hosts showed plenty of endeavour, but lacked quality in the final third and it was Sunderland who were in front with a goal from out of the blue after 31 minutes.
Palace were caught out when Fletcher rose to guide a fine header inside Julian Speroni's right-hand post after he was picked out by Van Aanholt.
The home side had another penalty shout when Wilfried Zaha went down under a challenge from Van Aanholt soon after going behind, but Dowd again saw nothing untoward.
Left-back Van Aanholt was carried off due to a shoulder injury sustained in that incident, with Brown replacing him.
Fletcher tested Speroni with a right-footed strike and Vergini was denied by the Palace goalkeeper in first-half stoppage time as Sunderland grew in confidence after going in front.
Palace made a bright start to the second half and they were level nine minutes after the break courtesy of Brown.
Zaha stood a ball up to the back post for an unmarked Chamakh and, although Pantilimon somehow kept out the striker's close-range header, the keeper was unable to prevent Brown from finding the back of his own net after Campbell had back-heeled the ball into his path.
Sunderland responded and Gomez fired home from 20 yards out to restore the visitors' lead, then Jedinak was shown a second yellow card four minutes from time after appearing to kick out at Gomez.
Fletcher added his side's third in time added on with a neat right-foot finish to add insult to injury for Palace.
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