Algerian Mahrez, one of the outstanding contenders for English football's player of the year, latched on to Jamie Vardy's 34th-minute cross to net his 16th league goal of the season.
Shinji Okazaki spurned a couple of chances early in the second half as the leaders attempted to ram home their advantage and Leicester, under severe pressure near the end, almost paid the price when Damian Delaney hit the bar in the dying seconds.
The defeat continued Palace's slide down the table after 13 league games without a victory and they remain the only team in the top four divisions of English football to have failed to have won a league game this year.
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(Reporting by Ian Chadband, editing by Ed Osmond)
