The world number 64 was down an early break in the opening set, but fought back to force a tiebreak and put himself in the lead.
The pair then traded breaks in the second, with the Argentinian eventually claiming the edge to draw level, but it was Norrie who came out on top in the decider to seal a 7-6 (5) 4-6 6-3 win.
Davis Cup
Great Britain through to quarters after 2-1 win over Czechs
The Briton will face Mackenzie McDonald in the quarter-finals.
Johanna Konta also claimed a spot in the last eight in Acapulco after defeating Varvara Flink in straight sets.
The number eight seed dominated from the off, breaking her opponent in the first game of the match, before clinching a 6-4 6-1 win in an hour and 23 minutes.
Number three seed Donna Vekic will be her opponent in the quarter-finals.
Britain’s Katie Boulter did not fare as well, as she was forced to retire from her second-round match at the same tournament.
The 22-year-old had been trailing Sonya Kenin 6-4 4-1.
Meanwhile, Kyle Edmund posted his first win of 2019 by beating Tommy Robredo in the second round of the Oracle Challenger Series at Indian Wells.
The British world number 27, who was given a bye in the first round, swept past his Spanish opponent 6-4 6-4 in an hour and 23 minutes.
The victory comes after a disappointing start to the year for Edmund, who fell to Tomas Berdych in the first round of the Australian Open last month.
Dan Evans, who narrowly missed out on his first ATP title at the weekend, claimed a 6-4 7-5 first-round win against Maxime Cressy.
In the women’s event, fellow Briton Heather Watson was dismissed by China’s Han Xinyun 6-4 7-6 (4).
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