The 18-year-old Konjuh will now meet Russian qualifier Anna Blinkova in the second round after she upset Latvia's Anastasija Sevastova 4-6, 6-1, 7-6 (10) after a tense deciding tiebreak.
Seventh-seeded Anastasia Pavlyuchenkova, a former champion in Moscow, is also in the second round after winning 7-6 (2), 5-7, 7-5 against Czech qualifier Katerina Siniakova.
Pavlyuchenkova's reward is a second-round match again former world No. 1 Jelena Jankovic of Serbia, who beat Latvia's Jelena Ostapenko 6-3, 6-3 in a match featuring 15 breaks of serve.
''It was very difficult for me to hold my serve because my opponent was returning very well and she was at times making so many great winners,'' Jankovic said.
Australia's Daria Gavrilova reached the second round with a 6-2, 6-2 win over Lucie Safarova, and goes on to meet third-seeded Carla Suarez Navarro of Spain in the second round.
In the men's event, sixth-seeded Pablo Carrena Busta of Spain came back from a set down to reach round two with a 3-6, 6-3, 7-6 (2) victory over Austrian veteran Jurgen Melzer. There was also a win for seventh-seeded Paolo Lorenzi, who saw off fellow Italian Federico Gaio 6-4, 6-4.