Watch: Gutsy Murray out of Australian Open
The outpouring of support seemed to lift him and despite grimacing with pain after some rallies he was nevertheless moving relatively freely as the Spaniard took the opening set in 50 minutes.
The in-form Bautista Agut had beaten world number one Novak Djokovic on his way to winning the Qatar Open at the start of the month and he broke again for 4-2 in the second as a hampered Murray began to struggle.
But Murray would not lie down. Watched by mum Judy Murray and brother Jamie he pulled off some vintage magic to win the opening point of the third set with an outrageous half-volley lob over the advancing Spaniard, followed by a deft drop-volley winner at the net.
Bautista Agut broke for 2-1 and just as the end looked like it would come quickly, Murray dug deep and had the crowd on their feet after a winner down the line at the end of long rally to break back.
"Right, let's go," Murray shouted and held to love with an searing ace to nudge ahead 3-2 and bring another mighty roar from the crowd.
Murray forced a...