Despite enjoying a 13-0 head-to-head record against Kohlschreiber coming into the contest, Federer said his mindset ahead of the deciding third set was to somehow avoid an early defeat against the German.
"Just somehow get through. Somehow try not to lose," the 37-year-old said.
"I'm very pleased I was able to find a way. I actually played a really good third set."
The 20-times Grand Slam champion, who has been stuck on 99 titles since triumphing at his hometown Basel tournament last October, next faces Fernando Verdasco after the world number 32 got past Italian qualifier Thomas Fabbiano 3-6 6-3 6-2.
"I think he (Verdasco) has got one of those wonderful forehands on tour. We've seen the great shot-making he has been able to produce over the course of his career," Federer added.
"I'm happy he's still going. He's a good player. He can definitely peak against the top guys. That's why he's still playing, for those kinds of matches, so I have to be careful."
The second seeded Swiss has never lost to Verdasco in six meetings, last beating the 35-year-old Spaniard in Dubai en route to the title four years ago.
Federer's task in the tournament became a bit easier with seventh seed Milos Raonic and fourth seed Karen Khachanov, both in the same half of the draw as the Swiss, suffering opening round exits.
Russian Khachanov was felled by world number 19 Nikoloz Basilashvili who prevailed 6-4 6-1, while Raonic was beaten 6-4 5-7 6-4 by German Jan-Lennard Struff.
Defending champion Roberto Bautista Agut later defeated wildcard Ramkumar Ramanathan 6-4 6-3.
Top seed Kei Nishikori is in action on Tuesday with an opening round clash against Frenchman Benoit Paire while third seed Marin Cilic meets Gael Monfils.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Pritha Sarkar)
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