The world No.85 had to tough it out before prevailing 6-3 7-6 (7-5) over Michael Mmoh to reach the round of 16 at the ATP-250 event on Wednesday (AEDT).
Walton's reward is a meeting with second seed Frances Tiafoe with a spot in the quarter-finals up for grabs.
It was Australia's only success as Rinky Hijikata, No.6 seed Jordan Thompson and James Duckworth all bowed out.
The 24-year-old Hijikata suffered a 6-2 4-6 6-3 loss to Chris Eubanks in his first outing at the opening claycourt event of the season.
Earlier, Duckworth was beaten 6-2 6-2 by qualifier Colton Smith and Thompson ousted 4-6 6-4 6-3 by wildcard Ethan Quinn.
Eubanks will play either eighth seed Kei Nishikori of Japan or Mitchell Krueger next.
Hijikata, who was bundled out by Novak Djokovic last week in round two of the Miami Open after second-round eliminations on hardcourt events in Phoenix and Indian Wells, was broken in the second and eighth games to lose the opening set.
The 2023 Australian Open doubles champion responded with two breaks of serve to claim the second set, but he cracked in the sixth game of the decider before eventually going down in one hour 57 minutes.
Top seed Tommy Paul opens his campaign against Chile's Cristian Garin on Thursday.
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