The British number three overcame an uncertain start, surviving three break points in his opening service game before eventually being broken by his Czech opponent on his second.
But Edmund, 22, then clicked into gear as he broke Vesely twice in quick succession to rattle through the rest of the set in just 38 minutes.
The Yorkshireman continued to make life miserable for the world number 62 in the second set as a battery of outstanding groundstrokes were sent back over the net.
And, following back-to-back breaks of serve, the Briton wrapped up the match 6-3, 6-2 in one hour and seven minutes.
The win sets up a second-round meeting with fourth seed and Wimbledon runner-up Cilic on Tuesday, when British number two Aljaz Bedene is in first-round action against Paolo Lorenzi.