The British No 1 was dragged into a deciding set by Anderson before he eventually sealed a 6-4 3-6 6-3 triumph, becoming only the 46th man in the Open era to reach the half century mark.
Standing inside the baseline, Murray took on his opponent???s booming serve from the opening game, and seized an early break through Anderson???s over-hit forehand.
But the Scot???s own serve was severely tested at 4-3 up, with Murray scampering in the heat to fend off break points, before holding with a forehand volley.
After serving out the first set, Murray???s focus appeared to waver following a harsh time violation and a double fault led to more break points, with Anderson snatching the second to move 2-0 ahead.
Murray???s mood was matching the boiling conditions and he tossed away another break, but immediately hauled one back with a couple of searing returns.
Anderson???s ferocious serving kept the Brit at bay as he made him wait further for a milestone win.
In a dramatic start to the decider, both men dropped serve, but Murray merely raised his intensity and bellowed with delight when Anderson???s wayward forehand gifted another break.??
Murray would make no further mistakes and Anderson dumped his backhand into the net to seal victory.