The Frenchman, ranked 19th in the world, was sent tumbling by Australian Matosevic, who was hammered 6-1 in the first set but won the next two on tie-breaks.
Gasquet served 13 aces but was unable to convert his service dominance, throwing away four break points in the second set before losing the tie breaks 7-5 and 7-3.
The Croatian-born 26-year-old moves into the third round, as Gasquet had been given a bye in the first.
Andy Roddick qualified for the second round after Sam Querrey picked up an injury in the first set of their match.
Sixth seed Roddick was leading 5-2 in the first set when fellow American Querrey was forced to withdraw with a back problem.
Roddick had broken his opponent twice and was serving for the set. He next plays France’s Jeremy Chardy.
Seventh seed Philipp Kohlschreiber of Germany also went through after he beat Go Soeda of Japan 6-4 7-5.
He is rewarded with a second-round match against British number three Jamie Baker.
Fifth seed Julien Benneteau went out, though, losing a lengthy encounter 6-3 2-6 7-6(4) to Taiwan’s Lu Yen-Hsun.
Lu will play American Ryan Harrison next.
Steve Darcis of Belgium overcame Australian hope Matthew Ebden 6-2 6-4 and will face Cypriot Marcos Baghdatis next, while Uzbekistan’s Denis Istomin pulled off a 6-4 6-3 win over Frenchman Michael Llodra to set up a clash with Spain’s Marcel Granollers.