Ten man Stanley lost out by the odd goal in five to local rivals Morecambe at the Wham Stadium.
The Reds had led through two goals from Billy Kee before Mark Hughes was red carded and the Shrimps fought back to take all three points.
It took just a couple of minutes for the chances to start, with Morecambe striker Stockton heading straight at Elliot Parish before Shay McCartan slalomed through the visiting defence but could only drive his shot narrowly wide.
Soon it was Stanley posing more of the threat, with Kee's slightly mis-hit shot pushed away by Barry Roche before a Conneely shot came back to the Stanley striker who could only find the side netting.
After 17 goals last season Kee was yet to get off the mark in the new campaign but wasn't to be denied a third time in the 26th minute. He turned his man superbly just outside the box and pulled the trigger, the shot looking to take a deflection to take it past Roche and make it 1-0.
John O'Sullivan was only prevented from adding a second by Roche's superb near post save moments later but soon they did have the two goal cushion.
This time Kee may have know less about it; O'Sullivan's cross seemed to hit the front man and divert into the left corner of the net.
Stanley's joy lasted just a minute as they were reduced to ten men. Stockton looked to turn Hughes a few yards outside the box, the defender stopped his progress and referee Michael Jones showed a straight red card.
In three minutes of added time the Shrimps pulled one back when Stockton rose highest to flick a Michael Rose cross beyond Parish and make it 2-1.
It would get worse for John Coleman's side who were pegged back a minute into the second half. A cross from the right ricocheted to Kevin Ellison, who drilled a right foot shot home to make it 2-2.
With the officials under the spotlight Stanley were denied a penalty when they felt Ryan Edwards had dragged down Matty Pearson before Kee's 30 yarder was saved by Roche.
Morecambe then should have been in front but for the prowess of Pearson. A poor back pass let the Shrimps in and the ball was squared for Stockton to shoot. He beat Parish but Pearson flung himself at the ball to block.
It wasa temporary reprieve, with Stockton cutting in from the left and firing right footed past Parish to turn things around and make it 3-2 to the visitors.
Stanley's attempts to level again brought shot wide from Sean McConville but Morecambe kept threatening with Ellison hammering over the top before Parish saved from Barkhuizen.
The ten men put on pressure late on with an Omar Beckles header beating Roche but sliding wide and a late melee unable to register a leveller. Instead Morecambe held on for their first Football League win at the Wham Stadium.
Stanley: Parish, Donacien, Beckles, Hughes, Pearson, O'Sullivan, Conneely, Hewitt, McConville, McCartan, Kee