The fifth seed found a crucial break of serve when leading the opener 5-4 to take the first set in Kazakhstan, before winning the next three games to put himself in the ascendancy in the second.
Despite a desperate last stand to stay in the match from Djere, Rublev held serve three times with relative ease to finish off his Serbian opponent.
Eighth seed Felix Auger-Aliassime served 14 aces to Roberto Bautista-Agut's two in their first-round match, but it was the Spaniard who advanced with a 6-4 7-6 (8-6) success.
Meanwhile, world number 10 Hubert Hurkacz overcame Francisco Cerundolo in straight sets.
Adrian Mannarino, the runner-up in Astana in 2020, came from a set down to beat Stan Wawrinka – playing as a wildcard – 1-6 6-2 6-3, and qualifier Zhizhen Zhang romped to victory with a superb third set to seal a 4-6 6-4 6-1 comeback win of his own over Aslan Karatsev.
At the Tokyo Open, fourth seed Frances Tiafoe cruised into the second round with a 6-3 6-4 triumph over Yasutaka Uchiyama, one of three Japanese players who crashed out at the first-round stage of their home tournament.
Uchiyama's countrymen Taro Daniel and Shintaro Mochizuki also tasted defeat at the hands of Bernabe Zapata Miralles and Brandon Nakashima, respectively.