Del Potro forced two breaks of serve in each set and saved all five chances the 97th-ranked Baghdatis held in the first set.
The eighth-seeded Dimitrov, who won his first career title at Stockholm last weekend, won 6-3, 6-2 against Alexandr Dolgopolov of Ukraine.
All six second-round matches on Thursday were straight-sets victories.
Vasek Pospisil of Canada overcame big-serving Ivo Karlovic 6-3, 6-4 to earn a quarterfinal against another Croat, Ivan Dodig.
Edouard Roger-Vasselin of France beat German qualifier Tobias Kamke 7-5, 6-3, and will play Daniel Brands of Germany, who beat Lukasz Kubot of Poland 6-2, 6-4. The winner is on course to meet del Potro in the semifinals.