Murphy - winner of this trophy in 2008 - was forced to halt a mini-revival from his fellow Englishman, who had chances to level at 3-3 and 4-4 before the 2005 world champion finally secured a meeting with his fellow former world champion Graeme Dott in the last 16.
A break of 54 handed Milkins the first frame only for Murphy to produce some of his very best stuff in moving 3-1 clear with runs of 68, 73 and 91.
Milkins continued to fight the good fight in contributing 50 to trail 3-2. It should have been 3-3, but the Gloucester player missed a red with a rest needing only one red and a colour to leave Murphy needing a snooker.
Murphy was not so wasteful as a clearance of 43 saw move him move 4-2 ahead. Milkins ran in a rapid 64 to close to 4-3 behind, but missing a red on an early run of 42 proved costly as his opponent returned to the table to win a safety exchange before sealing the frame for a 5-3 lead.
The world number four finished off the match in some style with a classy 85 illustrating his credentials as a possible winner of the event.
In the morning's other match, Hastings man Mark Davis completed a convincing 6-1 win over Cao Yupeng.
Davis was in a confident mood throughout the contest as knocks of 58, 50, 79, 68, 51 and 48 overwhelmed his young Chinese opponent.
He will meet John Higgins for a spot in the quarter-finals.
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