Medvedev took out Kyrgios in two hours, 58 minutes to pick up his ninth straight victory at a major and his first against the mercurial Aussie.
"You have tough opponents like Nick here", he said. "That is why we love tennis. Any match, anything can happen. I just want to play my best and try to run good, hit good shots and hopefully something big can happen."
Tsitsipas struggled, but eventually prevailed against Sebastian Baez 7-6 (1), 6-7 (5), 6-3, 6-4 in three hours, 22 minutes, saving all 12 break points he faced.
"I tried to go for my shots a bit more", the fourth seed said after the victory. "Certain moments I got a bit lucky with finding the right depth and coming in.
"On the return, I tried to stay as close to the baseline as possible which cut out time for him and push him aback a little bit more. I think those were some of the things I really tried to work on."
Rublev served 13 aces in a dominant 6-4, 6-2, 6-0 victory over Ricardas Berankis while Sinner rolled past Steve Johnson 6-2, 6-4, 6-3 to reach the third round in Melbourne for the first time.
Ninth seed Felix Auger-Aliassime survived four tiebreakers against Alejandro Davidovich Fokina 7-6 (4), 6-7 (4), 7-6 (5), 7-6 (4), Bautista Agut crushed Philipp Kohlschreiber 6-1, 6-0, 6-3 while 20th seed Taylor Fritz won an all-American affair against Frances Tiafoe 6-4, 6-3, 7-6 (5).
24th seed Daniel Evans received a walkover from Arthur Rinderknech, 27th seed Marin Cilic overcame qualifier Norbert Gombos 6-2, 6-3, 3-6, 7-6 (6) and 32nd seed Alex de Minaur rolled past Kamil Majchrzak 6-4, 6-4, 6-2. Dimitrov went down to Benoit Paire 6-4, 6-4, 6-7 (4), 7-6 (2).
Four matches involved two unseeded players as Maxime Cressy advanced past Tomas Machac 6-3, 1-6, 6-3, 7-6 (5), Taro Daniel downed Andy Murray 6-4, 6-4, 6-4, Pablo Andujar got by qualifier Alex Molcan 6-4, 7-6 (6), 0-6, 6-1 and Richard Gasquet retired trailing Botic van de Zandschulp 4-6, 6-0, 4-0.