Zenit lose 1-0 in Glasgow to the Scottish champions
Roberto Mancini's side take a one-goal deficit back to Stadium St. Petersburg for the second leg next Thursday.
The breakthrough for Celtic came in the 78th minute when McGregor, who had been a thorn in Zenit's side all night, thundered home a cross from the left after losing his marker at the far post.
Boih sides saw a number of bookings in the closing stages as they each pushed forward looking for a goal. Nothing was forthcoming and the game ended 1-0 to the hoops.
The second leg is at Stadium St. Petersburg on Thursday 22 February, kick-off 9pm.