Temperatures reached in excess of 30 Celsius (86 Fahrenheit) for most of the afternoon.
After a heat-enforced 10-minute break leading into the deciding set, Rogers dominated Bouchard to give American women a second victory Sunday in the first round.
Christina McHale also advanced with a 6-3, 6-3 victory over Italy's Sara Errani.
Defending champion and No. 1-ranked Angelique Kerber, who won the Australian Open and U.S. Open last year, and French Open winner Garbine Muguruza, highlight the women's draw in Brisbane.
Defending champion Milos Raonic and Rafael Nadal are featured in the men's draw.