Bertens, the first Dutchwoman to reach the last 16 since Michaella Krajicek, sister of former men's champion Richard, did so in 2007, followed up her win over Venus Williams in the third round with a display of power and poise.
Williams ended up winning against Giorgi, and will now face Julia Goerges in Thursday's semi-finals.
He will meet Rafael Nadal at a grand slam for the second time in a matter of weeks after the Spaniard eased past del Potro in the French Open semifinals en route to his 17th major.
Williams is back in action at Wimbledon following a hiatus from the tennis circuit after becoming a mother past year.
Giorgi went into the match having served the most double faults in the tournament and duly committed one in the first game, but recovered her composure to save it with a scintillating cross-court forehand.
Williams took the set in convincing fashion, 6-3, and yelled her trademark, "Come on!" as she clinched the final point.
"I'm just here just to be here, and to prove that I'm back", Williams added. "I'm really happy for you. This is all I care about".
Kerber took advantage of Kasatkina's erratic serving and errors to return to the last four.
But the American's formidable record could not dampen Goerges' jubilation on Tuesday because as she admitted: "After five years of first round losses, I have grown a lot".
Rapper friend Drake and fellow singer Justin Timberlake were among those rooting for her on Centre Court.
Goerges' win kept alive the possibility of Wimbledon staging an all-German women's final on Saturday as twice major victor Angelique Kerber takes on Latvia's Jelena Ostapenko in the other semi.
The 21-year-old is the only female player yet to drop a set in this year's tournament.