Konta nearly enjoyed the ideal start to her match, earning two break points in the opening game, but she failed to take either as Vekic also looked strong from the outset.
The world number one's 6-3 6-2 6-2 victory over Dustin Brown meant there was success for all four home singles players in action on Wednesday, with Murray's win following those of Johanna Konta, Heather Watson and Aljaz Bedene.
But the world number nine didn't even reach the last 16 as he was over-powered 6-4, 7-6 (7/3), 3-6, 6-3 by Spanish 18th seed Roberto Bautista Agut.
Andy Murray, Rafael Nadal and Venus Williams were made to sweat it out waiting on a sweltering day at the All England Club where players battled soaring temperatures and flying ants in their bid to make the last 32.
"And if I do, my coach said he'd shave his head".
Now Watson must try to win her first match in five attempts against Azarenka, who claimed the best win of her comeback so far 6-3 6-3 against Elena Vesnina.
The 27-year-old Azarenka produced the best performance to date of her comeback from maternity leave.
"I'm approaching this event like I am every other event".
Naomi Osaka bt Barbora Strycova (x22) 6-1, 0-6, 6-4.
He said: "It's obviously good to have more players playing in the slams and winning matches". Mini-breaks abounded before Konta edged ahead, needing just a single set point to close out a tight, high quality set.
"It's really fun that I'm able to go home and spend time with my son", the Belarusian said.
She took everything her Croatian opponent threw at her, and came back time and time again with answers to all the problems she encountered in a more-than three-hour battle.
The 26-year-old announced her intentions from the first game of the match when she twice drew Sakkari into the net before passing her with crisp cross-court winners to break serve. "She's definitely playing with confidence in the Slams". I just feel that it's sometimes an unfair comment, because we see a lot of players - I'm not going to name who - having conversations with their boxes. "I know how strong you are and you'll get through this!"
There are two all British doubles matches scheduled for Thursday.
Ostapenko, who turned 20 last month, had never won a tour-level title until her impressive run on the red clay at Roland Garros.