The Australian had to come from behind in their tough three-set battle, before prevailing 3-6 6-4 7-5 on Thursday.
Thompson, who served 12 aces to the American's 10, will now meet No.28 seed Giovanni Mpetshi Perricard for a place in the last 32.
With only eight ranking spots separating them, there was precious little between the two on court with Sydney-born world No.37 Thompson made to fight all the way for his victory.
Both players recorded a 63 first-serve percentage while Giron won marginally more points on first serve (74 per cent to 72).
Thompson, though won 69 per cent of points on his second serve as compared to Giron's 57 per cent, which proved to be the telling statistic.
Also through is Reilly Opelka, who earned his first ATP Masters 1000 win since 2022 with a 6-3 7-6 (7-4) triumph over American wild card Christopher Eubanks.
He will now face Indian Wells finalist Holger Rune in the second round.
China's Bu Yunchaokete beat Britain's Cameron Norrie 6-4 6-2 without facing a break point and qualifiers Jacob Fearnley and Thiago Agustin Tirante also advanced.
Former finalist Kei Nishikori withdrew from his Miami opener after reaching the semi-finals of an ATP Challenger Tour event last week.
"My body has not recovered enough from the matches last week. I will now go home and get ready for the clay court season," Nishikori wrote on X.
with Reuters
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