"Nick, last point was really borderline", Dumusois said after the world number 104 aced Kyrgios to make it 2-1 in the second set.
Four American players pulled off upsets at the Shanghai Masters on Monday. Basilashvili's victory over an under-the-weather del Potro at the weekend was just his second ATP title, the first coming in Hamburg in July.
Alex de Minaur starts his campaign on Tuesday against Vasek Pospisil.
Nick Kyrgios' effort is under question again after an umpire suggested a point in the second set of his Shanghai Masters defeat was "really borderline".
Woodbridge felt it was a pity that Kyrgios couldn't skip Shanghai - a compulsory Masters Series event - given it was the third successive year in which the Australian had gone out of the tournament in controversial fashion.
But he also sounded a note of caution.
Regardless, now ranked 38, Kyrgios will need to work hard to grab a seeding for next year's Australian Open in Melbourne. "You have no right to tell me that it's poor", he told the French official.
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Previous year in Shanghai, having received a warning for smashing two balls out of court in frustration, Kyrgios marched off midway through his first-round match against Steve Johnson.
Federer added: "He can win bigger tournaments and do all these things".
"But there is still a process in place that he needs to do like any other player needs to go through in order to be successful". Sanghai is not the only scene of a Kyrgios controversy.
Swedish umpire Mohamed Lahyani was reprimanded by U.S. Open organisers for going "beyond protocol" when he climbed down from his chair to give Nick Kyrgios a mid-match pep talk during his second round match against Pierre-Hugues Herbert.
The Australian, who at times appeared largely disinterested in his three-set loss to American qualifier Bradley Klahn, fired back telling chair umpire Damien Dumusois he "didn't care" for his opinion and "you have no right to tell me that it's poor".
A 22-time grand slam champion - in double and mixed doubles - and former Davis Cup coach, Woodbridge said Kyrgios should next year try to focus his energies on the majors.