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|Nick Kyrgios had to save a match point to down Kudla in the Cincinnati first round. (AFP/MATTHEW STOCKMAN)|
Last year's losing finalist Kyrgios finally saw off the plucky American on his sixth match point.
The Australian will next face Croatian Borna Coric, who beat Daniil Medvedev 6-2, 6-3.
But Kyrgios admitted that his chronic hip problems are starting to become a worry.
"This year has been tough. I started the year very well. Then obviously I hurt my elbow. Then I had an ongoing hip injury," he said.
"We have been definitely thinking about the options with my hip. There is only so much you can do before you have to, I guess, get surgery or something like that.
"Right now I'm just managing it. Obviously I'm feeling my right knee because I'm favoring my left side of my body a little bit.
"I'm just trying to get through it. The physio has been helping me. I've been doing rehab every day. I'm doing everything I can.
"As long as I can serve, I have a good chance to win still, I guess."
The 23-year-old Kyrgios has struggled since Wimbledon, retiring from his second-round match against Cameron Norrie in Atlanta before losing at the first hurdle to Stan Wawrinka in last week's Rogers Cup.
But the world number 18 dug deep against Kudla after losing a first-set tie-break, saving three break points in game 11 of the second set, before levelling the match on his third set point and winning a thrilling deciding breaker.
"It's always good to get a win at a tournament that you previously did pretty well in. I'm struggling a little bit physically. I'm not 100 percent. So I'm just happy to get through," Kyrgios said.
"Obviously I'm still able to serve at a high level, which is half the game. So I gave myself a chance today.
"I don't usually feel too many things on a tennis court, but I felt a little bit of adrenaline in the third set, which was good."
Weekend Toronto finalist Stefanos Tsitsipas ran out of gas in a 7-5, 6-3 loss to Belgian 11th seed David Goffin.
South Korea's Chung Hyeon rallied past a struggling Jack Sock for a 2-6, 6-1, 6-2 victory.
Chung handed the slumping American an eighth straight defeat, with Sock's last win coming in the opening round at Rome three months ago.
The 22-year-old Korean, who pulled out of Toronto last week with injury, prevailed after Sock took a long medical timeout in the second set for an apparent back problem.
Sock discussed the situation with officials before opting to play on, but Chung, an Australian Open semi-finalist earlier this year, charged through the remainder of the match to set up a showdown with fourth seed Juan Martin del Potro.
Unseeded Milos Raonic advanced past Serb Dusan Lajovic 6-3, 6-3 at the combined ATP and WTA event, the last major tune-up prior to the US Open on August 27.
Russian Karen Khachanov, who reached the last four in Toronto last week, put out Spain's Albert Ramos-Vinolas 2-6, 6-3, 6-2.
In the women's draw, Czech ninth seed Karolina Pliskova made a personal breakthrough as she beat Agnieszka Radwanska for the first time after seven previous straight-sets losses to the Pole, winning 6-3, 6-3.
"It was a tough round for me because I've never beaten her, never even taken a set," said Pliskova. "I was so happy when I won the first set because I thought, 'Finally, one set for me!'"
Australian 16th seed Ashleigh Barty defeated Czech teenager Marketa Vondrousova 6-3, 7-5, while 14th-ranked Elise Mertens topped Magdalena Rybarikova 6-4, 6-2.
Maria Sakkari beat Indian Wells champion Naomi Osaka 6-3, 7-6 (10/8).