Tsitsipas, still dealing with an ongoing shoulder issue and playing his first match of the Miami fortnight following a bye and walkover, respectively, in the opening two rounds, had 12 aces and saved three of four break points in the two-hour match.
"I was waiting for a very long time to get out there and play, it almost felt like a vacation this last week staying in Miami so I am glad got started," said Tsitsipas.
"Difficult match, I won't lie, it was difficult out there against an opponent that has shown good tennis against top players in the past."
The match remained on serve until Tsitsipas broke for a 5-3 lead in the opening frame before going on to hold at love to grab the first set without facing a single break point.
Tsitsipas was tested early in the second set when he fell behind 0-40 and needed to win five consecutive points to hold serve and draw level at 1-1 but could not repeat that escape while serving at 4-5 as Garin broke to love to force a decider.
The Greek squandered two break points chances in the opening game of a third set but got the break he would need at 4-4 when Garin double-faulted while at 40-30 in a game the Chilean went on to lose before Tsitsipas served out the match.
Up next for Tsitsipas will be Russian 14th seed Karen Khachanov, who advanced to the round of 16 with a 6-2, 6-4 win over Czech Jiri Lehecka earlier on Monday.
Tsitsipas has a 6-0 record in head-to-head meetings against Khachanov with their most recent coming at the Australian Open semi-finals in January where the Greek prevailed in four sets to reach his first Melbourne final.
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