Gavrilova broke Mertens early in the match and clinched the opening set with ease. She continued to attack her rival in the second set but world number 36 Mertens saved a break point and changed her gameplan to counter the Australian.
"I started to be more aggressive, stepped up to the baseline and came to the net more," Mertens said. "I made her play more defensive, so I'm happy with my game in the end.
"I've worked a lot on my offence. Tennis at the top is so aggressive, so direct, that's the style I want to play."
Mertens reeled off three games in a row to level the contest at one set all and used the momentum to prevail in the decider.
Goffin, who beat world number four Alexander Zverev in his opening match, struck 27 winners in his 6-4 6-2 victory over Kokkinakis to clinch the tie. Belgium then won the mixed doubles dead rubber.
"I'm feeling so well on the court since I arrived here in Australia," Goffin said. "I'm playing really well, I've won two good matches with a high level, so I'm very happy how I've started the year."
Belgium can reach the final if they beat Canada in their next match and Australia fail to better their result against Germany.
Earlier, Angelique Kerber beat Eugenie Bouchard 6-1 6-3 while Alexander Zverev defeated Vasek Pospisil 6-4 6-2 as Germany sealed a comfortable 3-0 victory over Canada.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Christian Radnedge)
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