Ram and Salisbury, the fourth seeds, captured their second Grand Slam title after last year's Australian Open. They were Australian Open runners-up this year in their only other Slam final.
"To have won this with Rajeev is amazing," Salisbury said. "It's a dream come true.
"The last three years, he has been an incredible partner. I couldn't ask for anyone better to be on the court beside me. We look forward to hopefully getting some more titles together."
The winning duo, semi-finalists at Wimbledon in July, shared a $660,000 (558,492 euros) top prize.
"I can't ask for a better partner," Ram said. "It has just been the most unbelievable ride and we're not going to stop now."
Murray and Soares, seeded seventh, missed a chance for their third Slam crown after the 2016 US and Australian Opens.
Soares won last year's US Open double crown alongside Mate Pavic