Popyrin lost 7-6 (7-5) 6-7 (5-7) 5-7 after a see-saw encounter that could easily have gone the other way.
The Australian, himself ranked 19, will have the consolation of a very good workout ahead of the US Open, and may appreciate some rest after such a draining encounter. However, his chances of a top-16 ranking and better seeding slot in New York appear to be gone.
When Rublev's passing shot beat him to seal victory, Popyrin looked very weary before smiling ruefully as he dragged himself to the net to give his opponent a hug.
"I was trying to fight, to keep believing. In the end, somehow out of nowhere at 6-5, 30-0 for him, I make amazing returns. I was able to break him," Rublev told the Tennis Channel.
"Both of us deserved to win."
Another Aussie ousted in round three was Adam Walton.
The 26-year-old Queenslander was beaten 7-6 (7-5) 7-6 (7-3) by Czech Jiri Lehecka, who is drawn to take on either No.5 seed Ben Shelton or Roberto Bautista Agut in the last 16.
Elsewhere on Tuesday, Spanish world No.2 Carlos Alcaraz recorded a 6-4 6-4 win over Serb Hamad Medjedovic.
The Monte Carlo and Rome champion secured his 13th consecutive win at an ATP Masters 1000 tournament, and next faces Italian Luca Nardi as he continues for an eighth Masters 1000 title.
"I know he (Medjedovic) doesn't like to run too much from side to side, so my plan was to make him run as much as I could," said Alcaraz.
"I'm just glad that I did it most of the time, tried to defend in a good way ... I'm just really happy to get the win."
Karen Khachanov also enjoyed a straight-sets victory, the Russian 14th seed overcoming American Jenson Brooksby 6-3 6-3 to set up a meeting with third seed Alexander Zverev or Brandon Nakashima.
That clash was stopped with the German up 6-4 5-4 and serving for the match when heavy rain halted play for the day.
Earlier, Popyrin and Rublev dropped their opening service games.
The first set thereafter went per script to the tiebreak. Popyrin blasted an ace at 5-5 then the Russian, facing set point, double-faulted.
The second set went with serve to the breaker. After double-faulting at 4-5 Popyrin broke for 5-5, only for Rublev to sweep the next two points and level the match.
The contest had begun shortly after 11am local time, in fierce sunshine, but for some time humidity had been rising and clouds building. Suddenly, with the third set poised at 2-1 to Rublev, a cloudburst flooded the court and sent players and spectators rushing for cover.
Upon resumption, the match went with serve until, at 30-30 and 5-6 down, the Australian had to step away due to a ringing mobile phone. It may have made no difference but his subsequent serves weren't quite there and Rublev pounced.
The former world No.5 now meets Argentina's Francisco Comensana, who beat Reilly Opelka 6-7 (4-7) 6-4 7-5.
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