Garath McCleary's cross from the right side of the pitch in the 92nd minute found Austin alone in the box. His powerful header beat Canadian goalkeeper Kenny Stamatopoulos and sent Jamaica to its first win of the tournament.
"We did not play well in the second half but it is a good performance, we fought a lot," said Canada head coach Benito Floro. "We worked a lot for the 85 minutes."
Canada, which opened with a 0-0 draw against El Salvador on Wednesday, was unfortunate to surrender its first goal of the tournament moments away from earning a deserved point.
The Canadians (0-1-1) wrap up their preliminary round Tuesday against Costa Rica at Toronto's BMO Field. Canada, ranked 103rd in the world, will need a big result over the 41st-ranked Ticos in order to move on to the next round.
"It just comes down to us getting a win against Costa Rica and the one advantage we have is that we're playing at home," said midfielder Julian de Guzman, who returned to action after missing Canada's opening match against El Savador with an injury.
"Going into Canada will give us confidence to get a win. It's still there for us, I know the guys won't feel happy about the result (against Jamaica) and it's just a matter of bouncing back."
Jamaica (1-0-1) took the provisional lead in Group B with the win. El Salvador earned a 1-1 draw with Costa Rica in Saturday's other Group B match. El Salvador and Costa Rica are both 0-0-2, though Costa Rica owns a two-goal differential.
The Gold Cup is the regional soccer championships for teams in North America, South America and the Caribbean.