City got off to the perfect start with a goal after just 64 seconds when a mis-hit cross from on-loan Arsenal starlet Kerrea Gilbert flew into the far corner of wrong-footed Owain Fon Williams' net.
But City's joy was short-lived as Stockport hit back with a leveller after five minutes. The Leicester defence failed to clear to allow former Fox Chris O'Grady to hook home from 18 yards.
Full-back Gilbert almost grabbed his second on 14 minutes, but this time the keeper was alert to the danger and clawed away the ball from beneath the bar.
The home side kept up the tempo with Fon Williams forced to use his feet to keep out a close-range prod from Steve Howard.
City continued to dominate but with the Stockport defence having composed itself the interval arrived with no further significant goalmouth drama.
The second-half began as a tepid affair until Stockport came to life in the 66th minute when substitute Greg Tansey fired a 20-yard free-kick just over.
Alarm bells were ringing for City soon after when another substitute, Oli Johnson, found space in front of goal but David Stockdale was on hand with a brave block before James Tunnicliffe drove straight at the City keeper.
Stockdale, making his debut for City after joining on loan from Premier League side Fulham, saved his best until the last minute with an instinctive save to keep out Gary Dicker's goalbound shot.