Frenchman Gaston raced through the first set 6-2.
But 27-year-old wildcard Jasika created six break-point opportunities - none of which he was able to convert - in the second game of the second set before play was suspended due to heavy rain and thunder on Sunday with the pair locked at deuce.
Omar Jasika had the crowd on his side on Court 3. (James Ross/AAP PHOTOS)
Organisers confirmed there would be no further action on the outside courts until at least 1700 (AEDT), but play was continuing under closed roofs on the three main show courts.
Jasika shot to prominence in 2014 when he became the first player in 28 years to win the boys' singles and doubles titles at the US Open.
The Australian has won only one of his four previous main-draw grand slam singles matches - and that was way back at the 2016 Australian Open.
Gaston lost his only previous clash with a local at Melbourne Park, falling to  Christopher O'Connell in the opening round in 2022.
Hugo Gaston has made a fast start against Omar Jasika at Melbourne Park. (James Ross/AAP PHOTOS)
Jasika is one of five Australians in action on day one at Melbourne Park along with Adam Walton, Aleksandar Vukic, Li Tu and Daria Saville.
Two-time defending women's champion Aryna Sabalenka from Belarus begins her title defence on Rod Laver Arena on Sunday evening against American Sloane Stephens, the 2017 US Open title-winner,
Other big names in action on day one include men's No.2 seed Alexander Zverev from Germany and sixth-seeded Norwegian Casper Ruud.