The Austrian outfit had gone into the night sitting top of the tightest group in the competition ahead of Chelsea and AC Milan, but their dropped points mean that Chelsea now lead the way courtesy of their win at San Siro.
Salzburg had looked on course to pick up their second win of the group stage when they broke the deadlock after 14 minutes, with Nicolas Seiwald firing into the bottom corner after a quick break from the visitors.
However, it was Zagreb who created the better chances on the night and they may feel more disappointed with only coming away from the match with a point.
Having already seen one goal chalked off due to a foul in the buildup, it looked as though they would be denied again when Robert Ljubicic's strike just before half time was initially ruled out due to an offside.
However, VAR intervened and ruled that the player in question was not interfering in play, allowing the equaliser to stand.
A string of presentable chances then went begging for the hosts in the second half, although Salzburg came within inches of a winner themselves when Luka Sucic struck the crossbar 15 minutes from time.
The crossbar at the other end then came to their rescue in second-half stoppage time, though, as Petar Bockaj's free kick from 25 yards crashed against the woodwork with the keeper beaten.
A share of the spoils ultimately leaves Salzburg second in the group, although they are now two points clear of Milan and Zagreb in third and fourth respectively.