The 38-year-old Peruvian, in his third spell at the club, came on as a substitute in the 76th and made an impact three minutes later, turning in the shot by Robert Bauer with his hip.
Leverkusen, who had Wendell sent off in the 90th, wasted a stoppage time penalty when Omer Toprak's weak effort was saved by Felix Wiedwald.
Werder, unbeaten in their last four matches, are 14th with 26 points. Leverkusen lose further ground to the European spots and are 10th on 31.
Korkut had been hoping for a winning debut after taking over this week following the departure of Roger Schmidt, who was sacked following last week's 6-2 demolition by Borussia Dortmund.
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The hosts, who take on Atletico Madrid in their Champions League Round of 16 second leg next week, grabbed a sixth minute lead when Kevin Volland headed in after the ball had bounced off the crossbar.
Complacent Leverkusen then inexplicably yielded control to Werder who piled on the pressure after the break.
They came close with two chances early in the second half before Pizarro, the Bundesliga's all-time foreign top scorer, struck his first league goal since May last year.
Leaders Bayern Munich, seven points clear at the top, take on Eintracht Frankfurt, who have lost their four previous league games, on Saturday.
Second-placed RB Leipzig host struggling VfL Wolfsburg.
(Reporting by Karolos Grohmann, editing by Pritha Sarkar)
