Dimitrov needed six match points to complete the victory which maintains his challenge for a debut at the season-ending championships in London next month.
He is currently sixth in the Race to London rankings with the top eight qualifying. Rafael Nadal, Roger Federer, Dominic Thiem and Alexander Zverev have already claimed their spots.
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"It's nice to get through that match in straight sets," Dimitrov said. "I knew it was not going to be easy but at the same time I thought I kept a very good intensity throughout."
South Africa's Kevin Anderson, another player with hopes of qualifying for London, plays later against Spaniard Fernando Verdasco.
Results from the Stockholm Open Men's Singles Quarterfinals
4-Juan Martin del Potro (Argentina) beat 7-Yuichi Sugita (Japan) 6-2 7-6(6)
1-Grigor Dimitrov (Bulgaria) beat 5-Mischa Zverev (Germany) 6-3 6-4
6-Fabio Fognini (Italy) beat 3-Jack Sock (U.S.) 6-7(3) 7-6(2) 7-5
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