Watts – who is on loan at Home Park until the end of the season – slotted home his side’s second of the evening, adding to Pantuche Camara’s opener to put Argyle 2-0 up just after the half hour mark.
But after the break, Orient mounted a comeback at Brisbane Road, with Louis Dennis pulling one back ten minutes after the restart.
And after veteran Jobi McAnuff’s leveller with quarter of an hour remaining, Watt’s side were stunned when former Gateshead frontman Danny Johnson netted a stoppage-time winner to send the hosts through to the third round.
It was a tough result to take for 20-year-old Watts – who played the full 90 minutes – and his team, who missed out on an eyecatching meeting with Premier League side Tottenham Hotspur in the next round.
The only other Magpies loanee who could have featured in midweek was Morecambe stopper Jake Turner, but the 21-year-old was rested as the Shrimpers set up a third round meeting with Newcastle thanks to a 1-0 victory over fellow League Two outfit Oldham Athletic.