Battling Rublev puts Russia on brink of Davis Cup semi-final
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"There's a lot of pressure when you know that you have to win and your opponent has nothing to lose," said Rublev.
World number two and US Open champion Daniil Medvedev can wrap up a place in the last-four when he faces Elias Ymer's brother Mikael later Thursday.
If 94th-ranked Mikael pulls off a shock result, the quarter-final will be decided by the doubles rubber between Aslan Karatsev and Rublev against Andre Goransson and Robert Lindstedt.
Russia are looking to reach a second successive semifinal while Sweden's last appearance in the final four was in 2007.
The winner of the tie will face Germany for a place in Sunday's final with Novak Djokovic's Serbia meeting neighbours Croatia.