Emulating fellow Canadian Milos Raonic, Pospisil fired 23 aces en route to a 6-3, 6-4, 1-6, 6-3 win over Italy's Fognini.
Pospisil broke Fognini three times in the match and converted on 83 per cent of his first serve points.
Fognini converted two of five break points, with both coming in the third set.
Pospisil will face James Ward in the third round.
Britain's Ward advanced to the main draw as a lucky loser, but has played well since. He advanced to the third round with a 6-2, 7-6 (4), 3-6, 6-3 win over Jiri Vesely of the Czech Republic.