Rune – the fifth seed for the tournament in Bulgaria – required two tie-breaks to clinch victory in his first meeting with Van Rijthoven on the ATP Tour, eventually claiming a 7-6 (7-2) 7-6 (8-6) win.
After the match, Rune expressed his relief at avoiding a third set, saying: "I didn't want to go to three sets. If it [had happened], I was going to fight for sure and try to take it in three, but I'm very happy to finish in two."
Meanwhile, eighth seed Oscar Otte beat Dragos Nicolae Madaras 6-3 6-2, and Portugal's Nuno Borges overcame Mirza Basic in three sets to tee up a meeting with defending champion and top seed Jannik Sinner.
In Tel Aviv, seeds Aslan Karatsev and Tallon Griekspoor fell at the first hurdle with three-set reverses to Tomas Martin Etcheverry and Vasek Pospisil, respectively.
Etcheverry rallied after Karatsev took a second-set tie-break in comprehensive fashion to post a 6-2 6-7 (0-7) 6-4 win, while lucky loser Pospisil stunned Griekspoor for a 4-6 7-6 (7-3) 7-5 triumph.
Fifth seed Botic van de Zandschulp fared better, however, dispatching Joao Sousa 6-2 6-3 after producing a top-class display of serving.
The Dutchman will meet Britain's Liam Broady in the next round after he recorded a straight-sets win over Hamad Medjedovic.