Kerber, ranked second in the world, ran out 6-1, 6-3 winner in Leipzig, to make it 1-1 in the rubber, in her first match since beating Serena Williams in last Saturday's final in Melbourne to claim her first Grand Slam title.
The 28-year-old German admitted she was still feeling the effects of her Melbourne triumph after an endless round of media interviews on her triumphant return.
"I felt the tiredness in the second set, but the spectators here helped carry me through," she said after needing 74 minutes to see off Bacsinszky, ranked 15th in the world.
The Swiss had taken an early lead after Kerber's team-mate Andrea Petkovic lost to teenager Belinda Bencic in the opening rubber.
The 18-year-old Bencic, ranked 11th, enjoyed a 6-3, 6-4 win.
There are three more rubbers to come on Sunday when Kerber plays Bencic and Petkovic faces Bacsinszky before the doubled rubber when Martina Hingis will play for the Swiss.