At the outset, Mertens appeared set for a dominant win after racing to a 4-0 lead in the first set but Martic fought back with a combination of powerful groundstrokes and wily sliced backhands.
Facing a break point while serving at 4-0, Mertens made a double fault, which started a frustrating four-game skid for the Belgian.
Martic forced a tiebreak and was trailing 2-3 when Mertens switched gears, eventually clinching the opening set with a sharp forehand winner.
With momentum on her side, Mertens was more clinical in the second set as she converted three out of four break point opportunities to close the match in under two hours.
Mertens, ranked 37th in the world, extended her winning streak to eight matches after defending her title at the Hobart International earlier this month.
"It's been amazing, the good start of the year, no words to describe this feeling," Mertens, who was unable to participate in the Australian Open qualifying last year as she was still involved in the Hobart International, said in her on-court interview.
"First time in the main draw here, defended my Hobart title... it's amazing. If you have a strong belief you can do something, you can get there."
The 22-year-old ended up winning at Hobart last year for her first WTA title and finished the season in the top 40.
She will face either fourth seed Elina Svitolina from Ukraine or Czech qualifier Denisa Allertova in the quarter-finals.
(Reporting by Hardik Vyas in Bengaluru; editing by Sudipto Ganguly)
