The World number one beat Croatian Donna Vekic 6-3 4-6 7-6 (7-4) at the Porsche Tennis Grand Prix.
Osaka will now face Anett Kontaveit, who was able to progress after Victoria Azarenka retired.
Petra Kvitova will face Kiki Bertens in the other semi-final.
UAE beats Canada in T20 World Cup thriller
The UAE defeated Canada by five wickets in a thrilling Twenty20 World Cup Group D clash in Delhi, chasing down a target of 151 with two balls to spare.
Zimbabwe stun Australia to stay unbeaten in T20 World Cup
Australia were dealt an early blow in their quest to reach the Super Eight stage of the Twenty20 World Cup after the depleted former champions crashed to a 23-run defeat by Zimbabwe in Colombo on Friday.
Arsenal miss chance to go six points clear with draw at Brentford
Arsenal has wasted the chance to restore their six-point lead at the top of the Premier League table as they were held to a 1-1 draw at Brentford on Thursday, with Noni Madueke's second-half header cancelled out by Keane Lewis-Potter.
India thump Namibia ahead of Pakistan clash
Defending champions India continued their march towards the Super Eight stage of the Twenty20 World Cup with a 93-run win over Namibia in a group A contest at Arun Jaitley Stadium on Thursday.
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