After a goalless but entertaining first half on a heavy Hard Lane surface, Cousins broke the deadlock on the hour with a stunning 25-yard strike.
Two minutes later the midfielder, back in the side due to Owen Mountain's absence, converted substitute Karl Fitzgerald's cross for 2-0.
Reduced to 10 men by injury to Chris Frost, Easington allowed their hosts a foothold 10 minutes from time when Simon Toseland reduced arrears.
But Fitzgerald confirmed the points were heading back to East Yorkshire when he thumped home an injury-time volley following Paul Morrill's fine pass.
The win lifts the Eastenders into the top six and sets up a tasty encounter at home to leaders Church Warsop Miners Welfare this weekend.
Easington: C Holden, C Smalley (K Fitzgerald, 61), A Barnett, M Hutchinson, S Campbell (A Graham, 68), J Cousins, C Frost, P Morrill, P Farndale, G Thurkettle (J Purse, 65), C Havercroft.