Murray and Broady sealed a 6-3 7-6 (5) win in Roehampton, sealing victory via a tie-break.
The Scot has limited himself to doubles action this week, teaming up with 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 results also included Dan Evans beating Aidan McHugh 7-5 6-2, Heather Watson seeing off Beth Grey 6-0 6-2 , Raducanu defeating Broady 6-3 6-3, and Harriet Dart winning 6-4 7-6 (5) against Jodie Burrage.
Additional reporting from Press Association