The No.7 seeded Australian won 6-1 6-1 in just 68 minutes to set up a last-eight showdown with third seed Ben Shelton.
Thompson, who will improve on his career-best 43 when the next rankings are released, broke Kudla's serve five times during a dominant display in Texas.
The 29-year-old Sydneysider also fired down seven aces to his opponent's zero, won 76 per cent of points on his first ball and hit 21 winners.
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Into his 19th ATP-level quarter-final and fourth in Dallas, it follows an impressive start to the season for the No.3 Aussie who defeated Ugo Humbert and Rafael Nadal in reaching the semi-finals in Brisbane.
Earlier on Thursday (Friday AEDT), compatriot Rinky Hijikata suffered a 6-3 7-6 (7-5) round-of-16 defeat to eighth-seeded German Dominik Koepfer, who next meets second seed Tommy Paul.
In the night session, top seed Frances Tiafoe advanced with a 6-4 6-3 victory over 19-year-old fellow American Alex Michelsen.
The 26-year-old Tiafoe closed out his opening match of the tournament on his first match point when Michelsen, who has risen into the top 100 over the past year, netted a forehand for Tiafoe's fifth service break.
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"I wanted to kind of be the aggressor," Tiafoe said.
"I played really well, I thought. A lot of creativity, and I got a little lucky at times."
The world No.14, who lost in the second round of the Australian Open after reaching the US Open quarters last summer, will play American Marcos Giron in the last eight.
Frenchman Adrian Mannarino will take on Aussie James Duckworth in the fourth quarter-final after Yoshihito Nishioka of Japan withdrew prior to their clash due to illness.
With AP.
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