Murray, a two-time winner of the Cincinnati Masters, won 6-4, 6-4 and faces ninth seed Hubert Hurkacz or Alejandro Davidovich Fokina next.
But British Men’s number one Dan Evans lost 6-2, 4-6, 6-3 to Diego Schwartzman.
In the women’s draw, Heather Watson progressed with a 6-2, 3-6, 6-1 victory over Belarusian Aliaksandra Sasnovich.
Murray, a three-time Grand Slam champion and winner of this event in 2008 and 2011, pulled out of the Olympic Singles in Tokyo with a minor thigh strain.
The 34-year-old has been added to the US Open main draw – where he won his first Grand Slam title in 2012 – which begins on August 30. (BBC)