In one of the best matches of the tournament so far, which lasted three hours 10 minutes, Vekic's resilience kept her constantly in with a chance of an upset but sixth-seeded Konta's big serve frequently pulled her out of trouble.
Vekic, ranked 58th in the world, beat Konta in the final of last month's WTA tournament in Nottingham and the pair embraced at the end of a gruelling contest in the afternoon heat.
"It is a nice feeling not having to keep going out there," Konta said. "We were out there for a long time and both of us battled incredibly hard and whoever was going to draw the (short) straw was going to be hurting and obviously I feel very fortunate to have got through that."
Konta, aiming to be the first British woman to win the Wimbledon women's singles title since Virginia Wade in 1977, will face Greece's Maria Sakkari, ranked 101st in the world, in the third round.
Konta has reached the third round for the first time at the All England Club, but after a strong year in which she has established herself in the top 10 the Briton will be looking to advance much further.
She edged the first set, winning the tiebreak to four points, but then the 21-year-old Vekic broke serve in the fifth game of the second set to get the upper hand and the Croat served out to force a deciding set.
The Australia-born Konta saved break point to move into a 7-6 lead but had to regain her composure after taking a tumble in the next game as Vekic held serve to keep the contest level.
With Konta leading 9-8 the Croat saved match point with a blistering ace, but a big forehand winner from the local hope created the second match point and the duel was over when Vekic's return found the net.
(Reporting by Simon Evans; Editing by Jon Boyle and Ken Ferris)
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