Richard Walker and an own goal restored parity before another defensive mistake saw Mace regain the lead at the break.
We started slowly again and that’s something we need to look at.
But Andy Rosenberg struck after the restart and Sidle took his tally to four games in two games to complete the recovery late on.
AFC Flixton joint-boss Stuart Delmonte said: "We started slowly again and that’s something we need to look at.
"A better side in this league would have punished us. Despite giving ourselves a mountain to climb, we showed character and passion to win this game. The players deserve big credit for that.
"But we realise we won’t get away with our sloppy start again."