Blue Square Bet Premier side Alfreton took the lead early on and laid siege to Orient's goal having pulled it back to 3-2 - striking woodwork four times in total - but the League One side clung on.
The hosts flew out of the blocks in Sunday's second-round tie and were ahead on four minutes when top-scorer Paul Clayton fired under Ryan Allsop from 12 yards.
Ben Tomlinson then struck a post and Josh Law's deflected free-kick hit Allsop's bar as the non-league side dominated but Dean Cox levelled for Orient against the run of play on 25 minutes.
And on the half-hour mark, the visitors were ahead when Lee Cook's corner was flicked on and Mooney - who scored the goal that broke Gloucester's resistance in the previous round - volleyed home.
Alfreton were shell-shocked and Mooney had his second just six minutes later, heading home Cook's left-wing cross.
On 52 minutes Alfreton pulled a goal back as Tomlinson scrambled home from close range and while the hosts peppered Allsop's goal - Law hitting the bar with another free-kick and Tomlinson from open play - Cox grabbed another late on as Orient booked a third-round trip to Hull City.