The unseeded Australian whipped the American 6-2 6-2 and never looked to be in any bother on Friday as he maintained control from start to finish, even though the match was suspended late in the first set from the day before.
Thompson dished out seven aces to Eubanks' two. The world No.71 also won 86 per cent of his points on first serve and 80 per cent on his second, while converting four out of nine break points.
Such was his domination the Sydneysider won 60 points in the match compared to Eubanks' 34 and was never broken, as the local favourite conjured up just one break point opportunity but couldn't convert.
Next up for Thompson is top-seed Taylor Fritz, who needed three sets and three hours to get past veteran Andy Murray 6-7 (7-2) 6-3 6-4 on Friday.
The Fritz-Murray match, pushed back from Thursday because of rain, was also briefly interrupted by climate protesters and he will need to get back out on court later in the day for his meeting with Thompson.
Fritz, a 25-year-old from California who is the top-ranked American man at No.9, saved all seven break points he faced in the third set against Murray, the 36-year-old from Britain who owns three grand slam titles earned before he had two hip operations.
There was a trio of break chances in the match's final game, which Murray led love-40 before Fritz got out of trouble.
"I feel like when I'm down those break points, sometimes the opponent might feel some added pressure to it," said Fritz, who hit 17 aces with tennis balls he called "awful."
He managed to serve it out for his 40th victory of the season; only No.1 Carlos Alcaraz and No.3 Daniil Medvedev have more.
Fritz had failed to come through in a similar situation late in the opening set, which he led 5-3 and held one set point on Murray's serve there. Fritz then had two more set points on his own serve in the following game, but could not close the deal.
With the temperature around 30 Celsius and 60 per cent humidity, Fritz was scheduled to return to face Thompson for a spot in the semi - after a minimum of 90 minutes of rest, as required by ATP Tour rules.
"I can't say I've experienced this - playing three hours with the heaviest balls in the world in humid conditions and having to turn around and play again," Fritz said.
Murray said he couldn't remember the last time he played twice in one day.
"It's pretty rare that it happens," he said. "There is not necessarily, like, a strategy for that. Because it's so rare, you just try and focus on the first match.
"And then if you get through it, try and rest and recover as much as you can before going out there. But it's far from ideal."
No. 2 seed Frances Tiafoe and qualifier Shang Juncheng finally got on court for their fourth-rounder.
Tiafoe won 6-2 6-3 in 72 minutes, and was going to return later to meet No.9 Dan Evans in the quarter-finals. Evans defeated Alexander Shevchenko 6-4 6-3.
No.5 seed Grigor Dimitrov had a much simpler day. He moved into the semi-finals when No.13 Ugo Humbert withdrew before their match with an injured left leg.
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