Orient, led by former Glovers boss Russell Slade, took the lead after 30 minutes through Mooney. But before that the hosts had efforts from Darren Purse and Dominic Blizzard cleared off the line.
Orient's lead lasted just six minutes before Kieran Agard scored following an Ed Upson free-kick.
Two minutes later Yeovil were ahead when an Upson corner was headed in by Bondz N'Gala.
The second half was a story of two penalties. After 62 minutes a Blizzard shot was handled but Yeovil's skipper Paul Wotton fired wide.
Orient rode their luck and in the 87th minute Mooney was brought down by goalkeeper Marek Stech and got up to hammer the spot-kick home.