It is the second year in a row that Yorkshire players have scooped both awards. Bairstow, 26, succeeds Adam Lyth, having topped the Championship averages with 1,108 runs at 92.33 to earn an England recall and help his county retain the title.
Leaning, 21, succeeds Alex Lees, having made 902 runs at 40.08, including three centuries in the first half of the season. He is the fifth Yorkshire player in the past nine years to claim the award, after Adil Rashid (2007), Bairstow (2011), Joe Root (2012) and Lees.
Other previous winners of the award, which was first made in 1950, include Fred Trueman (1952), Geoffrey Boycott (1963), Alastair Cook (2005), Stuart Broad (2006) and Ben Stokes (2013).
The CWC's Peter Smith Award, for services to the presentation of cricket to the public, was awarded posthumously this year in honour of Richie Benaud. It was received on behalf of Benaud, who passed away in April, by last year's winner David Lloyd.
The CWC Book Award went to Stephen Chalke for Summer's Crown, The Story of Cricket's County Championship.