Medvedev makes winning return after Australian Open final loss
Elsewhere 18-year-old Jakub Mensik beat Borna Coric 48 hours after becoming the youngest finalist at a tour event since 2021, in Doha.
The man who ended the Czech teenager's dream run in Qatar on Sunday, Karen Khachanov, was also among the round of 32 winners.
World number four Medvedev got his title defence at the hardcourt ATP 500 event up and running, overcoming Shevchenko 6-3, 7-5.
The Russian top seed shrugged off his month-long absence - part holiday, part injury - breaking his Kazakh opponent in three consecutive return games in the first set before reeling off four games in a row from 3-5 in the second set to complete an 88-minute triumph.
In January, Medvedev reached his second Australian Open final. He lost to Rafael Nadal in the Melbourne decider two years ago, and this time had to settle for second best in the season's opening Grand Slam to Jannik Sinner.
He was forced to skip his title defence in Doha last week because he was dealing with a number of physical issues –- with his foot, adductor, and shoulder.
"In general, to be honest, I am happy with my level, because it is not easy to come back after an injury.
"You always try to straight away put your game back, but its not easy and I'm happy to beat such a good opponent and am looking forward to the next round," said Medvedev courtside.
He added: "When you take a week or two off, but it's something you planned, then it's kind of easy tennis-wise, just a couple of points or games to get back.
"I did take a week and a half off by myself and then when I came back I started feeling pain here and there, not practising 100 per cent.
"It's very tricky, because when you come into a match you have to forget about it and when you think about something other than tennis you are going to lose the match."
Medvedev will meet next Lorenzo Sonego after the Italian despatched wild card Sumit Nagal of India 6-4, 5-7, 6-1.
Mensik let slip a match point and then saved one in his 4-6, 6-3, 7-6 (9/7) victory over Coric to book a date next with Spain's world number 24 Alejandro Davidovich Fokina.