But Konta lost 2-6 7-6 7-5 to Vekic, who she now faces in the second round of Wimbledon.
He faces Russian 30th seed Karen Khachanov next.
"Maybe I will have a chance in that she will feel more pressure than me, especially on Centre Court. I'm having a great season". It's just a nice feeling not to keep going out there.
Andy Murray made light work of crowd-pleaser Dustin Brown to cap a brilliant day for British players at Wimbledon. This was simply not a part of the script.
She hopes to become become the first mother to win the Wimbledon singles since Evonne Goolagong Cawley in 1980.
By this point Konta and Vekic were tiring and the third set could never have been as fast as the first two were. Konta played her patient game and before anyone would go for a refreshment or two, the first set was almost over and done with at 6-2. Some good volleying of her own and an errant return from the Croat brings the set level at 2-2. The Vekic match, when she was back-to-the-wall a couple of times and got herself out of sticky situations, you saw the determination, but also the calmness that hasn't always been there.
"Jo has the potential, but it will take a little bit of time". The nerves were settled as Konta takes the last three points to hold and lead 7-6.
"That was strange.it was in my nose and in my ear", he said.
Bernard Tomic's comments post his Wimbledon exit continues to be be a key topic of conversation, especially in the 24-year-old's home country.
Murray said it showed what could be achieved by the nation's sports stars.
"He's a very talented kid. A lot of other tour players thought we were so well looked after, and we weren't". He came to quite a few of my matches.
Frenchman Jo-Wilfried Tsonga sailed into the third round with a 6-1, 7-5, 6-2 win over Simone Bolelli of Italy.
Johanna Konta admits she is finding the Wimbledon experience "draining" but insists she has the stamina to mount a serious challenge for the title at SW19. She could be heading for the second week at Wimbledon for the first time. I don't have much to lose. "It's done", she said. United States 24th seed Sam Querrey and Britain's Aljaz Bedene also won their second round matches.
Belarusian Azarenka is competing in her first Slam since giving birth to her first child Leo in December. That's just me, though. Vekic was looking just too sharp and forced her way to two set points.
And if I do, my coach said he'd shave his head.
"You get a sense that is something she has worked incredibly hard on, the discipline to be focused on each point in the routine she has talked about and it proved itself in one of her biggest moments here", she said. "You can shave your legs." .
In reaching the quarter-finals of the Australian Open in January, Konta enjoyed a relatively serene path, winning each round in straight sets.
French Open champion Jelena Ostapenko joined them in the third round with a late evening three-set win over Canadian qualifier Francoise Abanda.