Zahavi scored twice in the first half, before Emre Kilinc pulled one back for the Super Lig runners-up on the stroke of half-time.
But Mario Gotze added another, with the Israeli rounding out his hat-trick in the 84th minute.
The German added the fifth minutes later to help the Dutch runners-up claim a strong lead ahead of the second leg.
PSV boss Rodger Schmidt said: It was a good evening for us. It was the first league game with fans in a year and a half so we wanted to show them good play.
“The energy of the fans came through in the football. We were constantly sharp, active and gave few chances away. We had the efficiency that we lacked last season.
Greek champions Olympiakos edged Neftci Baku 1-0 with Mohamed Camara netting the winner before being sent off.
Serbian champions Red Star Belgrade suffered a 2-1 defeat away to Kazak club Kairat.
Young Boys drew 0-0 at Slovan Bratislava, while Mura and Ludogorets also played out a goalless draw.
Polish champions Legia Warsaw won 2-1 over Flora with a 90th minute winner from Rafael Lopes.
Swedish champions Malmo FF got past HJK Helsinki 2-1.