The Eagles surged back to lead at the break thanks to Benteke's second and Marc Guehi's first goal for the club.
However, Cornet secured a point for the Clarets four minutes after the restart with a devastating left-footed volley for the Ivorian's fourth goal in as many league games.
While a draw is not enough to lift Burnley out of the relegation zone and above Leeds, who play on Sunday, it stretches their unbeaten run to four games before the visit of Tottenham on 28 November.
Palace, meanwhile, extended their own undefeated sequence to seven - five of which have been draws - even if they failed to leapfrog Manchester United into eighth.
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