Murray and Broady won 6-3 7-6 (5), triumphing via the tie-breaker after being broken at 5-4 up in the second set.
The three-time grand slam champion, only in doubles action this week, had also won twice alongside Lloyd Glasspool, against Dom Inglot and Alastair Gray and then Edmund and Joe Salisbury.
Murray told the BBC: “I thought we returned really well and Naomi was unbelievably solid up at the net. She cleaned up a lot of high balls.
“I feel like we deserved to win that match. We played better, we had the chance to close it out and at 6-3 5-4 we didn’t get it, but it is a good one to get through.”
Sunday’s victory, secured by a superb backspin volley from Broady, comes after Murray spoke about the prospect of playing in the US Open, which is scheduled to start behind closed doors at Flushing Meadow on August 31.
Murray, who has been hit significantly by injuries since 2017, said: “The situation I’ve been in last few years I’ve not had opportunity to play in many slams, I don’t know how many opportunities I’ll have left to play in slams.
“So, while I’m feeling relatively decent, obviously there is a risk there but I want to try and play in them and enjoy the biggest events again. I missed that a lot, I’ve missed it.
“I love playing the biggest events, even though this will be different with no fans but that is something I care about and willing to take a risk to go and play.”
The win for Murray and Broady left their Union Jacks team two points behind the British Bulldogs overall with one match to go.
The Bulldogs then secured a 63-56 victory thanks to Salisbury and Harriet Dart beating Jamie Murray and Heather Watson 6-4 6-2.
Earlier on Sunday, world number 44 Edmund beat Jan Choinski, ranked almost 300 places below him, 6-2 6-1.
Edmund, who broke Choinski four times and made just eight unforced errors, will now prepare for the US Open, and he said: “I want to get on the road again. The tournaments are scheduled so I hope that it stays that way and we can play some competitive tennis.
“This week has been nice as it’s probably the first time in a long time I’ve actually played some matches outdoors since Acapulco (the Mexican Open in late February). I feel like each day I’ve slowly improved so that’s been really good.
“It’s different conditions out in America of course but it’s nice to just get on the match court really.”
Sunday’s results also included Dan Evans beating Aidan McHugh 7-5 6-2, Watson seeing off Beth Grey 6-0 6-2, Raducanu defeating Broady 6-3 6-3 and Dart winning 6-4 7-6 (5) against Jodie Burrage.