Lions came back from a goal down to take maximum points in the closing stages of what was a grandstand finish.
Brady were forced to play an hour of the match with 10 men after Ben Kohler was sent off after 30 minutes, although manager Joel Nathan was "unsure what his first booking was for".
The visitors regrouped and took the lead 15 minutes into the second half via Mogilner but he was injured whilst scoring and had to come off.
Lions stepped up a gear and they drew level after some great football. This gave them the belief they could get a result and they capitalised on some sloppy defending to forge ahead. But Brady came back and won a penalty straight away which Charlie Kasler converted.
The Lions then capitalised on "a bad mistake" from Brady and from an attempted pass back to the keeper, Craig won the ball and slotted home the winner.
The game ended with Brady’s Jordan Nathan carried off with a nasty looking knee injury. Lions also finished with 10 men after goalkeeper Jordan Sharifian was sent off.
Stand-in manager Will Beresford told JC Sport: "Excellent team performance from Lions, much improved from last week."
Brady and boss Joel Nathan told JC Sport: "Unfortunately it was not our day today. We may have blown our league chances but still have great hopes in both cups and that starts next week when we travel to Manchester for a Morrison cup game."