The tie seemed set for extra time when Jonathan Lawson dispatched an 89th-minute penalty but Akinyimika netted right at the death to send the young Hornets through.
After a goalless first half, the match really came to life after the break.
Amin Nabizada, back in action for the junior side, opened the scoring after 64 minutes, before Kaheim Dixon levelled up for the visitors shortly afetrwards.
It was all going Charlton's way when Zach Mitchell volleyed in to make it 2-1 with less than 10 minutes on the clock, but that was only the start of the drama.
Lawson's penalty brought Watford back level, before Akinyimika won it to send the Hornets support at the Orbital Fasteners Stadium home happy.
The Hornets, also fourth in the Professional Development League, now progress to the last eight.
Watford XI: Macaulay; Eames, Mullins, Gorgiou, Kamga; Barrett (Akinyimika 84), Shevchenko (Smith 66), Keyes; Nabizada (Gorman 89), Lawson, Assiedou
Unused subs: Browne, Scriven
Forward Kash Odiase, meanwhile, has joined Walton & Hersham on loan for a month.
The 18-year-old links up with the Southern League Premier Division South leaders, hoping to make an impact in the seventh tier.