Dyson fired a 65 in the opening round on Saturday and needed only a 72 on Sunday to take the top spot at Glasgow Gailes with a 5-under-par 137 total. The 26-year-old, who will be making his third British Open appearance, was joined by Paul Bradshaw, Andrew Willey and Anthony Millar, who also earned their spots in the field at Glasgow Gailes.
At Irvine, Jonathan Cheetham secured the top position with rounds of 66-68. The 37-year-old, who competes on the Asian PGA Tour, finished one stroke clear of Sweden's Martin Erlandsson.
A group of five players finished knotted at 139 at Irvine, with Andrew Oldcorn and Sven Struver coming out of a playoff to earn their place in the Open.
Paul Wesselingh, Sean Whiffin, Steven Tiley and Lloyd Campbell tied at 137 on the Kintyre course at Turnberry and all four players made it through.
Three players earned a spot in the field through qualifying at Western Gallies: Daniel Sugrue, Ben Willman and Lewis Atkinson.
In other British Open news, Steve Elkington, who was part of the playoff at Muirfield in 2002, has withdrawn with a groin injury. Elkington, who qualified at Congressional, will be replaced by Jimmy Green.
The 133rd British Open Championship will be contested at Royal Troon from July 15-18.