Hearts' Steven Pressley was bundled over in a clumsy challenge by Noel Whelan, and Wyness stepped up to calmly send Ryan Esson the wrong way from the spot.
Up until that point chances have been few and far between, although Wyness had also rattled the bar seconds before the interval with a spectacular overhead effort.
Man-of-the-match Kevin McNaughton was only inches away with a 25-yard volley, but neither goalkeeper had a busy afternoon.
After the introduction of Steven Craig for Noel Whelan, the visitors' attack looked more lively.
Hearts felt they should have got a second spot-kick when Stephen Simmons was upended by Jamie Winter, but referee Mike McCurry wasn't interested.
Craig drew a fine save from Hearts' keeper Craig Gordon when he picked the ball up on the left side of the box and curled in a right-foot shot. Gordon was quickly off his line and pushed the ball round the post at full stretch.
But there was more controversy in the closing stages when Aberdeen thought they had equalised five minutes from time.
Michael Hart floated a delightful lob from 20 yards towards the top corner, and somehow Gordon got his fingertips on it to push the ball against the post.
Craig was on hand to tap home the rebound, but the linesman spotted an offside and the goal was disallowed to the fury of the large travelling support.
And in the last few seconds the Dons claimed a penalty when Christophe Berra slid in to rob Darren Mackie, but to be fair to the youngster it looked a good tackle.
Hearts, after an unbeaten January, were able to leapfrog Aberdeen into fourth place in the SPL table.