The 23-year-old Beck will meet 31-year-old Marcos Daniel of Brazil in the last four, with both hoping to reach his first career final.
The 73rd-ranked Daniel beat Florent Serra of France 6-4, 7-6 (1) in the quarterfinals.
Igor Andreev became the only seed to advance, as the No. 3 beat fifth-seeded Frenchman Jeremy Chardy 6-3, 6-3.
The 26-year-old Russian will next face Brazilian qualifier Thomaz Bellucci, whose sixth-seeded opponent Nicolas Kiefer retired with a foot injury when leading 6-3, 0-1.
Beck, currently at a career-high No. 51, proved tougher than Crivoi on the big points as both struggled to land barely half of their first serves.
The left-handed Beck forced four service breaks to his opponent's three on the clay surface.
Daniel trailed 4-1 in the opening set then reeled off five straight games for the set. He twice fell a break behind in the second but fought back each time to force Serra into the tiebreaker.
Saturday's schedule features two Brazilians in the last four of an ATP event for the first time since 2004, when Gustavo Kuerten and Flavio Saretta reached the semifinals at Vina del Mar, Chile.
The 21-year-old Bellucci is ranked 119 but upset top seed Stanislas Wawrinka in the second round, and was runner-up on clay at Costa do Sauipe, Brazil, in February.
Andreev, ranked 27, lost the Gstaad final a year ago but won the last of his three tour titles in 2005.
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