The British number three, ranked 67, struggled in the first set, but hit back impressively to win 1-6 6-3 6-3 in an hour and 43 minutes.
Evans was particularly dominant on his first serve, losing only seven points, and a single break in each of the last two sets was enough to topple the world number 37.
Dan Evans beaten by tearful Muller in Sydney final
Evans will next play either top seed Dominic Thiem or Portuguese qualifier Gastao Elias as he continues his build-up to the Australian Open.
There was defeat, though, for British number eight Brydan Klein in the second round of the ASB Classic in Auckland.
Klein posted an excellent victory over Facundo Bagnis in round one, a player ranked 269 places higher, but Joao Sousa proved too strong, the Portuguese winning 6-3 6-4.
Evans: British success 'really good' ahead of Australian Open