كتب : Hatem Maher
Croatia, who were already assured a place in the quarter-finals, maintained their 100% record in the prestigious tournament.
Klasnic, who made his first start in the tournament, scored what proved to be the winner seven minutes after restart.
The Werder Bremen striker sent a low; curling shot from inside the area went past Poland keeper Artur Boruc after being set-up by Danijel Pranjic.
Croatia finished on top of Group B with nine points, three ahead of second-placed Germany, who also booked their last-eight berth.
Slaven Bilic's men will face Turkey in the tournament's quarter-final after the latter finished as runners-up in Group A.
Austria and Poland were both eliminated after collecting just one point.