Sabalenka, who won her second Grand Slam title at Melbourne Park last month, looked on track for a routine victory after the Belarusian edged a tight opening set and grabbed an early break in the next but Vekic had other ideas.
The Croatian, who came into the contest having won five of her previous seven meetings with the world number two, was one point away from going 3-0 down in the second set but came up with a superb sliced winner from deep to draw level at deuce.
It proved to be the turning point of the match as Vekic won 12 of the next 13 games to secure the victory and set up a clash with Sorana Cirstea (33), who beat Veronika Kudermetova (26) 6-1, 6-4.
"I feel the conditions here don't fit me well at all. It's really tricky for me to compete here in Dubai. A really tricky court for me," Sabalenka said.
"The whole match I was leading, I was winning. I won the first set. I was up with break. I didn't feel like I was up. The level was so bad today from me.
"She also didn't play her best. At the end, she stepped in and started playing aggressively because she saw I'm not playing my best at all. That's why she came back from that score that easy, because it was just like who's more lucky?"
"At all times I didn't give up," said Vekic, who is ranked 31 in the world. "I didn't stop believing I could win. I just kept fighting.
"I told my coach I'm taking this week as half holidays. I can tell you I spent more time on the beach than on the court. Maybe that's a good way going forward."
Wimbledon champion Marketa Vondrousova (24) was made to work hard by American Peyton Stearns (22) before sealing a 6-1, 5-7, 6-2 victory, before in-form former French Open champion Jelena Ostapenko (26) eased past Swiss Lulu Sun (23) 6-3, 6-3.
Brisbane and Abu Dhabi champion Elena Rybakina (24) advanced to the round of 16 after former world number one Victoria Azarenka (34) was forced to retire with an injury before the third set after her Kazakh opponent levelled the match at 4-6, 6-2.
Eighth seed Maria Sakkari (28) snapped a run of three straight defeats with a 6-2, 6-4 win over American Emma Navarro (22) and will take on Jasmine Paolini (28), who beat Leylah Fernandez (21) 6-3, 6-4.
