Osaka was visibly frustrated and trailed 5-1 in the final set but she refused to give up and found her rhythm to break Vekic twice and prevent her from serving for the match.
In the tiebreaker, a confident Osaka upped her baseline game and had two early mini breaks before wrapping up the match in two hours and 18 minutes. An infuriated Vekic even smashed her racket after losing the match.
"I told myself I didn't want to have any regrets here," Osaka said. "I was stressed out when I went down 1-5... but this (comeback) was pretty good because I don't play really well on clay."
Earlier, world number three Petra Kvitova came back from a set down to beat Anastasija Sevastova 2-6 6-2 6-3 and move into the tournament's semi-finals for the third time in her career.
Sevastova broke Kvitova twice to take a 3-0 lead as the Czech struggled with her first serve and made a slew of unforced errors.
Kvitova fought back in the second set and won 10 straight points before levelling the match.
Kvitova took command in the final set and broke a visibly upset Sevastova to take a 3-1 lead before easing into the semis.
"In the first set I missed almost everything. I was pretty slow and she just couldn't miss," Kvitova said. "In the second set it was very important for me to stay on my serve and the chance to break her came."
Kvitova will face Kiki Bertens in the semi-finals after the Dutchwoman beat fourth seed Angelique Kerber 6-3 6-4 in a little over 75 minutes.
Anett Kontaveit also progressed to the semis, where she will face Osaka, after former world number one Victoria Azarenka was forced to retire in the third set of their quarter-final trailing 5-7 7-5 3-0.
Azarenka had just come off a Fed Cup semi-final in Australia and had eliminated defending champion Karolina Pliskova in the previous round.
(Reporting by Rohith Nair in Bengaluru, editing by Ed Osmond)
