Etcheverry had beaten the Briton in the first round of the Australian Open earlier this year, but Murray won 7-6 (7-0) 6-3 this time around.
The match resumed at 3-3 after being suspended because of rain on Friday.
Murray convincingly won a first set tie-break before one break in the second was enough to secure victory.
The 36-year-old produced a delightful backhand cross-court lob on the way to breaking Etcheverry in the sixth game of the second set.
Murray survived a break point in the next game and went 5-2 up with a hat-trick of aces.
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Serving for the set, Murray was 40-0 down before getting to deuce with a backhand that just caught the line.
He then secured victory in two hours 16 minutes when Etcheverry, ranked 30th in the world, sent a forehand long and wide on match point.
Overall, Murray served 10 aces, hit 28 winners compared to 17 from his opponent, and saved seven of seven break points.
He will play Tomas Machac, who is 60th in the men's rankings, of the Czech Republic in the last 32.
Murray is a two-time champion of the tournament but this is expected to be his final appearance in Miami as he plans to retire later this year.
"Miami has been like my tennis home really. I've done so much of my work and training and preparation here, and I love the city - and this is my last time playing here, so it's nice to get another opportunity," he said.
British number one Cameron Norrie, seeded 30th, is through to the third round after a 7-5 6-7 (4-7) 6-2 win over Italy's Flavio Cobolli.
Norrie will face defending champion Daniil Medvedev of Russia after he progressed to the last 32 with a 6-4 6-2 victory against Hungary's Marton Fucsovics.
Briton Jack Draper was knocked out as he suffered a defeat by Nicolas Jarry in three hours 33 minutes.
Draper failed to take two match points when serving at 5-4 in the decider before Chile's Jarry won 7-6 (7-5) 4-6 7-6 (7-2).
Only eight matches were completed on Friday because of rain and the start of Saturday's play was delayed by three hours because of the weather.
When play did resume, American Taylor Fritz was beaten 6-3 6-4 by Brazil's Thiago Seyboth Wild, while Denis Shapovalov of Canada earned a 6-2 6-4 win against Greece's Stefanos Tsitsipas.
Australian Open champion Jannik Sinner is through to the next round courtesy of a 6-3 6-4 win against fellow Italian Andrea Vavassori while top seed Carlos Alcaraz cruised past Spanish compatriot Roberto Carballes Baena 6-2, 6-1.
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