He pounced from 12 yards following Stephen Gleeson's nifty cross from the left.
Town had got themselves off to a great start with a goal after just four minutes.
It was a simple one too, as Colin Larkin crossed for strike partner Karl Hawley, and he stroked the ball home from ten yards.
Stockport responded well to their early setback though and Anthony Pilkington and Jim McNulty both missed decent half-chances.
The hosts began to dominate possession and Gary Dicker saw a 20-yard drive saved by Cobblers keeper Chris Dunn, but they couldn't find that killer final touch.
As Northampton tried to up their tempo Ryan Gilligan blazed a shot well over, then shortly before the interval Hawley wriggled free of Michael Raynes before seeing a shot blocked by County skipper Gareth Owen.
The visitors created the first opening of the second period when Stockport old boy Danny Jackman fired wide at the near post.
Stockport rallied again and Gleeson lifted a shot just over the crossbar, while on 66 minutes Pilkington turned sweetly before hitting a low effort right at Dunn.
The hosts finally got their equaliser with 15 minutes left when Gleeson raced down the left before crossing for youngster McSweeney, who blasted home his first league goal from 12 yards.
Cobblers skipper Chris Doig headed onto the crossbar late on, but in the end a draw was the least the hosts deserved.