After the first 12 games had gone emphatically with serve, the 19th seed fell behind by losing the first-set tie-break 7-4.
A solitary break of serve was enough to see Kohlschreiber claim the second set 6-3, but Paire then regained his one-set lead by winning the third by the same score.
Kohlschreiber won the fourth 6-2 at a canter and then quickly established a 4-2 lead in the deciding set.
Paire bravely broke back to force a winner-takes-all tie-break. However Kohlschreiber raised his game once again and blew the Frenchman away to prevail 7-4.
Kohlschreiber will face home favourite John Isner in round three.