Andy Murray won his first singles match since pulling out of the individual event at the Olympics as he beat France's Richard Gasquet in Cincinnati.
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 30 August.