Sabalenka won 2-6, 7-6(2), 6-4 in sweltering conditions in the German capital in the traditional Wimbledon warm-up event.
"I was thinking 'wow, that's how it is to play against me'," a smiling Sabalenka said of her opponent's early dominance, adding: "I thought it was her match for sure... I was just trying to stay in the game.
"I think it was a lucky match. I somehow, miraculously, came back in that second set."
Bartunkova, the world number 62, stormed to take the first set and led 4-0 in the second as a visibly frustrated Sabalenka struggled on serve and could not pose any kind of danger with her return.
With elimination looming, the Belarusian suddenly clicked into gear, winning five straight games before taking the second set in a tiebreak.
The heat began to wear on both players in the third set, with Sabalenka breaking her opponent three times but suffering two breaks herself before taking the match in two hours and 23 minutes.
Sabalenka will face American world number four Jessica Pegula, who beat close friend and podcast co-host Madison Keys 7-6(5), 7-6(8) earlier on Friday.
Pegula battles past Keys
In their first meeting since Pegula ended Keys' title defence at the Australian Open in January, both players each broke serve once in the opening set and forced a tiebreak, which Pegula won.
Pegula, a 2024 champion on the Berlin grass, ran out to a 4-2 lead in the second, but Keys fought back to once again set up a tiebreak.
Pegula triumphed 10-8 in the tiebreak to take the match and extend her winning run to seven without defeat over fellow Americans this calendar year.
Eighth seed Linda Noskova defeated Spanish wildcard Paula Badosa 6-1, 6-3 in Friday's second quarter-final.
Later on Friday, world number eight Elina Svitolina takes on Alexandra Eala, who eliminated Elena Rybakina on Thursday.
