Striker Freddy Eastwood inspired the win, scoring once and setting up the second as the Shrimpers won through 2-0 in extra-time.
It came after a goalless 90 minutes, in which Lincoln had enjoyed the best chances.
Eastwood smashed home the opening goal after 105 minutes, after a scramble in the area.
And five minutes later right-back Duncan Jupp birst forward alongside Eastwood to add a second on the counter attack