Murray endured a tough start to the tournament, with 31-year-old Delbonis challenging the former world No.1 in a gruelling first set.
Both players stubbonly defended their serve, with the first set not even featuring a break point.
The duo continued to hold serve in the resulting tie-break, until an unforced error from Delbonis was followed up with a double fault from the Argentine.
Murray enjoyed a much simpler second set, immediately breaking his opponent in the first game and then riding his momentum to secure a 7-6(4) 6-1 victory against the world No.36.
The Scot has not won consecutive matches since the Sydney Classic in January, and that'll be a hard pattern to change considering he faces world No.2 Daniil Medvedev in the second round.
Medvedev, fresh from losing his No.1 spot on Monday, received a bye in the first round in Miami. If he is to regain his top spot in the rankings, he'll first have to beat a three-time Grand Slam winner.
The pair have only faced each other one time in the past, with Medvedev coming out on top in Australia back in 2019.
And Murray was under no illusion just how tough his second round opponent will be.
When asked about the prospect of facing Medvedev, Murray said, "Obviously a tough match.
"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."
The duo are scheduled to take to the court on Saturday, March 26.