The Greek came through 4-6, 6-3, 7-6(8) after his 53rd-ranked Argentine opponent missed a match point in the 12th game of the deciding set.
Both men then saved more match points in the tie-breaker before Tsitsipas, a three-time runner-up at the Barcelona tournament, secured a ninth successive win following on from his Monte Carlo Masters title victory at the weekend.
Tsitsipas will face either 36th-ranked Arthur Fils (19) of France or Dusan Lajovic (26), the world number 59 from Serbia, in the semi-finals on Saturday.
Casper Ruud (25), the world number six from Norway who was runner-up to Tsitsipas in Monte Carlo, eased past Italian Matteo Arnaldi (23) 6-4, 6-3.
"It is not easy. You are playing for a spot in the semi-finals," said Ruud.
"I was able to come through in the end after a few crazy rallies."
Argentinian Tomas Etcheverry (24) also booked his place in the semi-finals with a hard-fought straight sets win over Britain's Cameron Norrie (28).
Etcheverry, ranked 30 in the world, one place higher Norrie, edged a tight match 7-6 (7-4), 7-6 (7-1) and will face Ruud for a place on the final.
"I spoke to my coach last night and also my team, that I have to play aggressively to win this match," the 24-year-old Etcheverry said.
"The tie-break was key. I played really good in that part of the match."
