City controlled much of the first half, and had a number of chances to wrap things up early on. But they lost their rhythm and direction, and Stockport did not quite have the edge to capitalise on this.
Stuart Elliott had a shot saved in the fifth minute, and seconds later, Ian Ashbee shot over from a great central position 20 yards out.
In the 10th minute, Stockport got their first shot on target after Luke Beckett did well to hold up the ball outside the area and shoot, but his effort was easily dealt with by Boaz Myhill.
City were enjoying by far the most of the possession, and they piled on the pressure with three corners in succession, which eventually culminated in Elliott powering a header just over the bar.
Stockport managed to compose themselves a little and the play became more even, though Elliot did try his luck with a wayward shot in the 20th minute.
Although City were very much in control, there was the odd sign of confusion in the defence, which nearly presented an opportunity to Rickie Lambert in the 24th minute, but he skewed his shot wide.
Elliott put a long-range shot wide on the half-hour mark, but there was no real bite from either side.
Four minutes later, there was a scare for City when Leon Cort slipped in the area and gifted Beckett the chance to shoot. He should have scored, but he hit a relatively tame shot at Myhill.
A minute later, Andrew Welsh tested City's keeper again, in Stockport's first sustained period of pressure, but it was another easy save. Lambert also saw a low long-range effort held by Myhill, as City let their guard slip a little.
The second half started as the first had ended, and it was arguably even quieter; neither team could manage anything really decisive. The home crowd livened up a bit when manager Peter Taylor took off ineffective strikers Danny Allsopp and Aaron Wilbraham for Stuart Green and Jason Price in the 60th minute.
But it was Stockport who had the best chance of the game when a clever flick from Beckett about eight yards out drew a superb save from Myhill to put it over the bar when it looked like a certain goal. Moments later, a low shot from Green whistled past the far post at the other end.
In the 82nd minute, Ashbee and Jim Goodwin were booked after a clash which sparked a brief melee on the pitch and prompted a rare exchange of chanting between the home and away fans.
Neil Cutler rescued the game at the death for Stockport with an excellent diving save to his left after Elliott had broken free and looked certain to win the Tigers all three points with an 18-yard drive.
However, results went against Stockport, and despite their hard work, they now find themselves propping up League One.