Muller, ranked 41, came through a marathon second-set tiebreak 18-16 having lost the opener and edged the deciding set tiebreaker 9-7 to seal an unlikely victory.
American Isner launched 43 aces but Muller, who banged down 26 of his own, refused to bow down. His escape was only one short of the ATP record for match points saved in a victory.
It was far more straightforward for top seed Andy Murray who eased past fellow Briton Aljaz Bedene 6-3 6-4.
The 29-year-old Murray required a single break of serve to take the opening set and recovered from a minor wobble in the second to subdue the Slovenia-born Bedene and book another all-British clash with wildcard Kyle Edmund.
Murray pounced in the eighth game of the opening set but his serene progress was halted when he surrendered an early break in the second set as Bedene capitalised on a momentary lapse.
With coach Ivan Lendl impassively watching on, world number two Murray quickly restored order and broke to lead 4-3 before rolling on to victory as he continues his build-up to Wimbledon.
"I thought it was not bad. A very different opponent from the first round. Aljaz plays most of the time from the back. I'm still getting used to the conditions a bit but I played well," Murray said on court.
His next opponent Edmund had earlier received a walkover when Frenchman Paul-Henri Mathieu pulled out before their match with a wrist injury.
(Reporting by Martyn Herman; Editing by Toby Chopra)
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