Rublev banked a 6-3, 6-4, 6-2 win on John Cain Arena -- his first of the year after two defeats in warm-up events, with either Australia's Max Purcell or Finland's Emil Ruusuvuori his next opponent.
"First of all I'm happy that I was able to get through in straight sets. It was super hot today so happy I can save some energy for future matches," said Rublev, who reached the quarter-finals at Melbourne in 2021 and won four titles last year.
"When you play against Dominic, it's never easy.
"We are really good friends and I know he is going through not easy times, so I just want to wish him all the best to come back at the same level that he belongs as fast as possible."
Thiem has shown encouraging signs on his return to the tour, rising from outside the top 350 to breach the top 100 this year after semi-final runs last year at Gstaad, Gijon and Antwerp.
He said a rib problem hampered him against Rublev.
"It happened at one serve in the second set. The sun was -- I was in a tough position to serve, so I completely changed the toss, and something happened after at the abs or at the ribs," he said.
Despite the defeat, Thiem said he remained confident of getting back to his best.
"Yes, that's the ultimate goal, and I hope that I can do it now during this season," he said.
"This tournament doesn't change anything because I just had a really tough opponent."
It was a seventh meeting between the pair but their first at a Grand Slam, with Rublev now leading 5-2.