Last year's beaten finalist took 68 minutes to beat the 27-year-old Inglot, a doubles specialist, 6-4 6-4.
Fognini is 17th in the world rankings and attempting to become the first Italian to finish the year inside the top-20 since Corrado Barazzutti in 1978.
It was a tougher day for the other seeds in action, Dmitry Tursanov and Denis Istomin.
Tursanov fought off veteran Marcos Baghdatis 7-6 (7/3) 7-6 (7/3) while Istomin, who made a run to the fourth round at the US Open, needed an hour and 45 minutes defeat Australia's Sam Groth 6-4 3-6 7-5.
Guillermo Garcia-Lopez beat Russian qualifier 6-4 6-1 but countryman Daniel Gimeno-Traver tripped up against Estonia's Jurgen Zopp 6-1 3-6 6-3.
Evgeny Donskoy was a 6-1 6-4 winner over Adrian Mannarino in the last singles match of the day.