Muller will retire at the end of the ATP Tour season but the 35-year-old Luxembourgish veteran showed he is not a spent force just yet after seeing off Marcos Baghdatis 7-6 (7-5) 6-3 in Newport.
Seeded sixth, Muller – a Wimbledon and US Open quarter-finalist – saved three of four break points faced and converted a pair of his own to progress to the second round on the grass in Rhode Island on Tuesday.
The 2016 Newport runner-up will face Marcel Granollers in the next round after the Spaniard defeated qualifier JC Aragone 7-6 (7-6) 6-0.
Harrison and De Minaur, meanwhile, fell at the first hurdle at the ATP 250 tournament.
American fifth seed Harrison was trailing Nicholas Mahut 2-0 in the second set having dropped the opener 6-2 before retiring.
Australian youngster De Minaur – the seventh seed – lost 6-7 (2-7) 7-6 (7-3) 7-5 against big-serving Croatian and 2016 champion Ivo Karlovic.
Bernard Tomic was beaten 6-3 1-6 6-2 by Dudi Sela, Ramkumar Ramanathan accounted for Victor Estrella Burgos 6-4 6-1, Sergiy Stakhovsky went down 6-4 3-6 6-3 to Jason Jung, while wildcard Christian Harrison outlasted Alex Bolt 6-3 4-6 7-5.
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