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The win gave them two points to reel in the British Bulldogs, and Andy's brother Jamie then helped to add three more.
Murray beats world No.10 Hurkacz to reach Vienna Open second round
Heather Watson and Jamie Murray teamed up against Liam Broad and Harriet Dart, taking three sets to win 6-3 3-6 17-15.
In the first Centre Court match of the day, Katie Boulter crushed Beth Grey with a straightforward 6-1 6-1 win.
Up next came Anton Matusevich against Kyle Edmund, with Edmund enjoying a similarly routine 6-2 6-1 victory.
Cameron Norrie lasted just four games before conceding defeat to rival Dan Evans in Saturday's Battle of the Brits.
Evans vs Norrie looked set to be the day's most competitive match-up, but Norrie was forced to retire due to injury after just four games, all won by his opponent.
Over on Court One at St James' Place, Jan Choinski needed three sets to negotiate his way past Alaistair Gray, 6-4 4-6 10-6.
Jodie Burrage also needed three sets to compete a 2-6 6-2 10-7 defeat of Emma Raducanu.
In the final match on the court, Dart and Beth Grey were in action against Alicia Barnett and Olivia Nicholls, with the British Bulldogs winning 6-2 6-2 to leave the two teams tied 45-45.
'I'm just better than them!' - Murray has no time for excuses
Union Jacks 45-45 British Bulldogs
K Boulter beat B Grey 6-1 6-1
K Edmund beat A Matusevich 6-2 6-1
D Evans beat C Norrie (retired) 4-0
J Choinski beat A Gray 6-4 4-6 10-6
J Burrage beat E Raducanu 2-6 6-2 10-7
H Dart and B Grey beat A Barnett and O Nicholls 6-2 6-2
A Murray and L Glasspool beat J Salisbury and K Edmund 4-6 6-3 10-4
J Marray and H Watson beat L Broad and H Dart 6-3 3-6 17-15
Murray aiming to be 'more clinical and ruthless' in Vienna
'My attitude was poor' - Murray admits to mistakes and bad decisions in defeat to Schwartzman