The forward – brought in from Portsmouth on a two-year deal – converted his first chance to set Leeds on their way to a third pre-season win.
Andy Gray and Ross McCormack were also among the goals but a robust Bodmin side ensured a competitive contest despite the huge gulf in divisions between the sides.
Leeds were without first-choice goalkeeper Paddy Kenny who was nursing a thigh strain and Luciano Becchio sat in the stands after twisting an ankle in Leeds’ 6-0 win at Tavistock. With Robert Snodgrass on the verge of a move to Norwich City, Varney started on the left wing.
He took just 17 minutes to make an impression as United’s steady pressure told. Michael Brown played Adam Drury into space and Varney ran in to glance his cross past former Barnsley keeper Kevin Miller.
Bodmin had produced the best effort of the game moments earlier when Shane Krac thumped a shot into the midriff of Jamie Ashdown in United’s goal and the hosts mustered several threatening attacks in the first half.
Adam Carter missed another opportunity when the ball fell to him four yards out and Ashdown tipped a strike from Sam Matthews over his crossbar.
Varney, however, should have scored again when he cleared Miller’s net with a volley from point-blank range and McCormack saw a diving header ruled offside.
Gray extended United’s advantage just before half-time when he pounced on Jason Pearce’s knockdown and McCormack converted a penalty on 45 minutes when Varney was fouled in the box. Aidy White then struck early in the second half following an error by Miller, completing the scoring in front of a crowd of 3,000.
United: Ashdown, Lees (Byram 70), Cranie, Pearce (Lewis Turner 78), Drury (Nathan Turner 68), White (Rogers 60), Green (Nunez 60), Brown (Thompson 60), Varney (Poleon 46), Gray, McCormack (Pugh 46).