The No 4 seeded pair, who collected titles in Indian Wells and Miami earlier this season, including a win over Azarenka and Barty in the Miami semifinals, once again defeated the pair with a 7-5, 7-5 margin.
It took Nadal 4 hours and 49 minutes to outlast Medvedev by 7-5, 6-3, 5-7, 4-6, 6-4 at the Arthur Ashe Stadium on Monday, ATP Tour reported.
The British-American pair stormed through the last five games of the opening set, with Murray knocking off a backhand volley winner to give them the early advantage.
At the Arthur Ashe Stadium in the US Open finals, Williams was denied a record-equaling 24th Slam title as the 19-year-old Andreescu on Saturday became the youngest Grand Slam winner since Maria Sharapova at the 2006 US Open.
Meanwhile, the Colombian pair have become just the third men's doubles team since 2003 to win both Wimbledon and the US Open in a single season.
"I started to play with more calm and aggression. I'm super happy to be back in the final of the US Open," said Nadal after making it to the final.
On Saturday, the 33-year-old will play Italian Matteo Berrettini, who had never made a Slam quarterfinal before this tournament. Berrettini beat Frenchman Gael Monfils in an epic five-setter earlier in the day.
On Tuesday night, Dimitrov came from two sets to one down and defeated the five-time US Open champion 3-6, 6-4, 3-6, 6-4, 6-2 in a match that lasted more than three hours.
Nadal will next face Argentine Diego Schwartzman, against whom he has won all seven of his FedEx ATP Head2Head meetings.
Wawrinka defeated a World No. 1 for the fifth time (5-21), and he is into the last eight at the season's final Grand Slam for the fifth time in his six most recent appearances.
