The Eagles scored twice inside the opening 15 minutes through Wilfried Zaha and James Tomkins, before Glenn Murray got a goal back.
Zaha restored Palace's two-goal advantage on 24 minutes but Izquierdo made it 3-2 ten minutes later. Albion dominated possession in the second half but were unable to take any of their opportunities and Izquierdo said: "It was a difficult game and in the beginning it was not the best performance.
"In the second half we had some opportunities but the result is because of the mistakes we made in the first half. It's a really sad day for us and it's a difficult moment we are in.
"We know it is a derby for us and the fans. It's a difficult day when you lose like this."
Brighton remain seven points clear of the Premier League relegation zone after Southampton lost 3-2 at home to Chelsea.
Albion entertain Tottenham at the Amex on Tuesday evening (7.45pm).
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