The sixth-ranked Greek was 5-3 down in the opening-set tiebreak but won four points in a row to take the set and followed up by breaking Lajovic's serve three times in a row at the start of the second.
That put Tsitsipas up against a clay specialist in the semis. Chilean player Christian Garin has two clay-court titles already in 2020 from back-to-back tournament wins in Cordoba and Rio de Janeiro in February, but he had to recover from a set down in his quarterfinal on Friday.
Garin beat Alexander Bublik of Kazakhstan 3-6, 6-4, 6-4 after responding to losing the first set by breaking Bublik in the opening game of the second.
The fifth-seeded Rublev produced a minor upset to eliminate fourth-seeded Roberto Bautista Agut 6-2, 7-5 after a topsy-turvy second set saw Bautista Agut twice recover from a break down.