Leigh may have hoped that Accrington's cup heroics against Bournemouth earlier in the week could have left them vulnerable this weekend. However, they got a nasty shock.
Leigh started the brighter getting some success down the left flank with Ian Fitzpatrick's runs, but their only shot was a left-foot curler from Wayne Maden that drifted well wide of the right-hand post.
Stanley struck in the 12th minute when a Peter Cavanagh cross was allowed to reach the far post and was prodded home at the second attempt by Andy Gouck from six yards.
However the weather then took control of the game. Swirling wind and rain disrupted the passing with the best goal attempt coming from a confident Gouck, whose 40-yard drive was well parried by Gary Kelly.
David McNiven and Carl Barrowclough worked hard up front for Leigh, but couldn't find any openings.
The deciding blow came in the second minute of extra time in the first half as a cross from the left was finished at the far post by Paul Howarth.
Leigh tried to get back into the game with a more forceful approach but were left open to attacks on the break. Steve Flitcroft had a one-on-one with the keeper that was squandered and Paul Mullin had a drive that was smothered.
The game became a stalemate until the final minute when McNiven grabbed a late consolation for Leigh.