The three-time grand slam winner secured a routine 6-2, 6-2 victory to set up a meeting with Roberto Bautista Agut next in Doha.
Murray had lost to Daniel in the second round of the Australian Open last month but was too good for New York-born right-hander on only their third meeting.
After a cagey opening, it was the Scot who clinched the first break in the sixth game to move 4-2 up having pushed his opponent hard in his previous two service games.
A second break sealed the opener after Murray managed to withstand a lengthy rally to pass Daniel at the net and take the first of three set points.
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World number 114 Daniel gained the upper hand at the start of the second with an immediate break but was pegged back instantly by the wildcard.
After Murray, a two-time winner of the competition held to love, he pressed home his advantage to break twice more and seal a straight-sets success in one hour and 20 minutes.
The 34-year-old will take on second seed and runner-up in last year's tournament Bautista Agut next.
It will be a first meeting between the duo since Murray lost to the Spaniard at the Australian Open in 2019, which many feared might be the final match of the double Wimbledon champion's career before he embarked on career-saving hip resurfacing surgery.