Former world number one Murray reached the last 16 of the event thanks to a 6-3 6-4 victory against third seed Gombos in Mallorca.
Murray dropped down to the Challenger Tour for the first time since 2005 in a bid to improve his fitness after hip resurfacing surgery in January, which threatened to end his stellar career.
The three-time grand slam champion started a singles comeback at ATP Masters 1000 tournament the Western & Southern Open before making a swift exit from the Winston-Salem Open.
Murray – who played a series of high-profile doubles tournaments after teaming up with Serena Williams in the mixed event at Wimbledon – then opted to skip the US Open in New York.
The 32-year-old turned to the Challenger Tour for the first time in 14 years, and after blitzing teenager Imran Sibille in his tournament opener, Murray saw off Gombos.
Gombos provided a much sterner test for Murray, who needed almost two hours to beat the Slovakian world number 115.
Murray will next face world number 240 Matteo Viola for a place in the quarter-finals.