Substitute Jeffrey Schlupp delivered the victor in the 76th minute to punish a lacklustre Bournemouth side who played with a one-man advantage from the 19th minute when Palace defender Mamadou Sakho was red-carded.
Zaha scored the first goal in first-half added time before being involved in an angry confrontation with Phil Bardsley, with Palace manager Roy Hodgson stepping in to prevent the fracas from escalating into something much uglier.
And the Eagles got the only goal of the game, with Jeffrey Schlupp taking the lead from left-back to drive the Eagles to his second goal in as many games.
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"It's always difficult playing with 10 men and then we lost our right back, so we were hard pushed", a happy Hodgson, interviewed at pitchside by former Liverpool striker Michael Owen and former Arsenal defender Lee Dixon, part of a sizeable pundit team, said.
While Palace are up to fifth, Eddie Howe's Bournemouth have won just one match in nine league games, losing four on the spin. "We needed a little bit more guile and creativity but it was lacking today".
Aaron Ramsdale should have done better with Schlupp's shot, though the Bournemouth goalkeeper made a vital stop from James McArthur's long-range effort to keep the cherries in contention.
As part of the coverage, viewers who subscribe to its Prime service can watch every one of this week's 15th round of games - the first time at least one game in a round of fixtures has not been screened live by a television channel.