The 39th-ranked Belgian beat the second-seeded Spanish star 7-5, 7-6 (7/4) in the third round to land the tournament's biggest upset yet, and will play either Tommy Paul or Valentin Vacherot next.
"It's a little disbelief and not feeling too much," Bergs, 27, said afterwards.
"You're just really in the moment, being focused on the points you have to play," he added. "I'll be very happy."
Alcaraz started with more quality until Bergs broke the Spaniard's serve to go up 3-2 in the first set.
The 23-year-old grabbed back that break in the next game but was uncharacteristically off-target, sending a mess of shots into the net.
He double-faulted to hand Bergs a critical break point in the 11th game, and lost the first set when a volley went short.
Alcaraz was visibly frustrated during the second set as he vented to his corner following a series of misfired shots that left him with no points in the fourth and eighth games.
Alcaraz forced a tie-break but could not reach a third set as Bergs closed things out just past the two-hour mark on his first match point.