Eighth seed Vekic has recorded wins over Sloane Stephens and Caroline Garcia since falling at the first hurdle at the US Open, but she was no match for Babos in a one-sided contest that lasted 67 minutes.
The Croatian’s surprising loss clears a path for top seed Julia Goerges - in action on Wednesday - to reach the last four.
Camila Giorgi, the only other seed involved on Tuesday, was tested by Pauline Parmentier before eventually emerging a 6-3 7-6 (7-4) winner.
The Italian will next meet qualifier Jil Teichmann, who beat Sorana Cirstea 6-4 6-2. Lucky loser Kristyna Pliskova also made the last 16, defeating Valentini Grammatikopoulou in straight sets.
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