The Scot battled to take the opening set in a tie-breaker before running away with the second in a 7-6 (4) 6-1 victory.
Neither player was able to gain an early advantage in a first set which did not feature a single break point.
The tie-breaker then carried on in a similar vein until Murray was able to take a 6-4 advantage on an unforced error from Delbonis, which the Argentinian followed up with a double fault.
The three-time grand slam champion went up an immediate break in the second and never looked like losing from there as his win set up a meeting with the Australian Open runner-up.
Speaking after the match, Murray said he was looking forward to the clash with Medvedev.
"Obviously a tough match," he said.
"He's played extremely well on the hard courts the last few seasons. He deserves to be right up there at the top of the game.
"It'll be a big challenge for me, it'll be a great test. I've got a big training block after this tournament, and it'll be a really good test for where my game's at and things I need to work on as well against him. So I'm looking forward to that."
A Brit is already in the third round after Cameron Norrie triumphed over Jack Draper in the all-British second round match. Norrie won in straight sets, 7-6 (5) 6-4, but Draper pushed him all the way.
"It's not easy against Jack. He's such a big talent and hits the ball big on both sides," said the 26-year-old world number 12, who also admitted pre-match nerves against the 146-ranked Draper.
He cast them aside to take the first set tie-break 7-5 with some aggressive net play after each had exchanged breaks of serve during the opening exchanges.
Norrie then saw off three break-points at the start of the second before going 2-0 up himself and went on to take the set 6-4 and line up a third round match with either John Isner or Hugo Gaston – a match which could further his aim of entering the world top ten.