Zaha has proven to be key to Palace's bid to remain in the top flight and he put the Eagles into an early lead when he tapped in at the back post after Luka Milivojevic's shot was parried into his path.
Roy Hodgson's team were in full control of the match, and a second goal arrived just before the quarter-hour mark when Tomkins forced the ball into the bottom corner from 10 yards after Brighton had initially made two blocks from goalbound efforts. The fireworks never quite materialised in a 0-0 league draw at the Amex Stadium last November and were certainly lacking when both sides fielded weakened teams for Brighton's 2-1 win in an FA Cup third-round tie in January, even if ex-Palace player Glenn Murray did raise questions over why VAR was not used on its debut as he got the hosts' late victor.
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Zaha restored the two-goal advantage with a bullet header from Milivojevic's ideal cross and, in a frenetic opening 45 minutes, Jose Izquierdo struck with a fine finish for the visitors.
Huddersfield is a point above Palace after Tom Ince's stoppage-time goal sealed a 1-0 victory over Watford. The Seagulls are in 13th place in the table and are seven points clear of relegation.
Chelsea scored three goals in eight minutes to beat Southampton, while Burnley kept their European hopes alive with a 2-1 victory over Leicester at Turf Moor.
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But with local bragging rights on the line, as well as precious points, Palace boss Hodgson knows the pressure that his players will be under after previously steering West Brom and Fulham to survival. A win could see them vault into 15th place, and a loss might have them fall into the relegation zone yet again.
The Clarets were soon two ahead as Kevin Long notched his first-ever league goal for the club, rising to convert Johann Berg Gudmundsson's deep corner.
Jeffrey Schlupp is also closing in on a return to action as Palace brace for a decisive final five fixtures.
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Burnley hung on to move within two points of sixth-placed Arsenal, with the Gunners travelling to Newcastle on Sunday.