Konta, 30, won the first set in an assured start before rain interrupted the second set in Ohio.
After the resumption, Czech Muchova, 24, claimed the second set in a close tie-break.
The world number 23 took the deciding set and will face former US Open champion Bianca Andreescu in round two.
In the men’s tournament, British number two Cameron Norrie could not convert a set’s lead to victory as American John Isner fought back to win 4-6, 7-6 (7-5), 6-4. Konta has had a troubled season, with no back-to-back wins and long injury breaks before she won the Nottingham Open this summer, her first title since winning the Miami Open in 2017. (BBC)