World number seven Goffin was beaten by Julien Benneteau in the second round in Melbourne, but, after a pair of Davis Cup singles victories in Belgium's tie with Hungary, was back on song as he saw off Simon in one hour and 24 minutes.
The top seeded Belgian dropped his first service game and twice failed to consolidate breaks before finally doing so at the third attempt to clinch the opening set.
There was no way back for Simon after he passed up two opportunities to serve out the first game of the second set, handing Goffin an advantage he would not relinquish.
Dustin Brown moved to within three points of victory over Nicolas Mahut with a one-handed backhand winner, but a back injury sustained in the process forced him to retire and sent the Frenchman into a second-round match against Jo-Wilfried Tsonga.
Richard Gasquet and Andrey Rublev advanced in straight sets, but eighth seed Yuichi Sugita went down 5-7 6-3 6-4 to John Millman, while there were wins for Ruben Bemelmans and Pierre-Hugues Herbert.
At the Sofia Open, fourth seed Philipp Kohlschreiber fell to qualifier Mirza Basic 7-5 7-6 (7-5) in the second round and Robin Haase – seeded fifth – lost his opening match to Marius Copil 7-6 (7-5) 6-4.
Marcos Baghdatis set up a second-round meeting with Adrian Mannarino by defeating Salvatore Caruso, while Jozef Kovalik joined fellow qualifier Basic in reaching the quarter-finals with a 4-6 6-2 7-5 victory over Lukas Lacko.
Elsewhere, world number 282 Roberto Quiroz delighted his home crowd when he finished off Yannick Hanfmann 7-6 (7-5) 6-3 in an Ecuador Open match held over from Tuesday due to a rain delay. His reward is a second-round clash with second seed Albert Ramos Vinolas.
Ivo Karlovic edged out Ernesto Escobedo 6-4 6-7 (6-8) 7-6 (7-4) and Alessandro Giannessi, Nicolas Jarry and Gerald Melzer also registered wins.