Meite won and converted an 87th minute penalty after being felled by home keeper Mike Cooper.
Meite then put the tie beyond Plymouth by firing home under Cooper in the first minute of stoppage time.
Target man Ryan Taylor claimed his third goal in two games to fire Argyle 1-0 up after 22 minutes.
Taylor tapped home from close range after strike partner Zak Rudden, on loan from Rangers, teed him up.
Reading levelled when impressive Josh Barrett – one of 11 changes to the team that won 2-0 at Huddersfield – cut in from the left and let fly with a thumping shot into the far corner with 35 minutes gone.
Barrett levelled again after 72 minutes when he cut in from the left to send a measured curling shot over a full stretch Cooper into the top corner.
Argyle could have gone 3-1 up but playmaker Danny Mayor's rising angled drive rattled the post and they were to rue that missed chance as Reading went on to win.