An 84th minute bullet header from Lee Frecklington set up a nervy finale for Martin Foyle's side, but some stout defending in the dying stages booked Vale a deserved place in the second round draw.
The Lincoln midfielder's last-gasp goal set nerves jangling in the Potteries and opened up some raw wounds inflicted days earlier in the capital.
The Valiants entered the match on the back of a grueling and ultimately demoralising midweek League Cup fourth round exit to Tottenham Hotspur.
Spurs forced an energy-sapping 120 minutes of football thanks to a late goal from Tom Huddlestone after Vale had led for 80 minutes.
The greatest domestic competition drew a paltry 3,884 to Vale Park, but those who stayed at home missed a dominant Port Vale conduct a relentless attack on Lincoln's goalmouth.
A five-goal advantage at the break would not have flattered the League One side and they would have had it were it not for a distinct lack of confidence from Vale's front men.
Sodje was the more culpable of the striking duo, spurning a succession a golden opportunities before making amends minutes before the half-time whistle.
The burly striker had a goalbound drive charged down on the 11th minute before Jeff Smith took meaningful aim on the Red Imps' increasingly vulnerable target. The winger's effort left Alan Marriott nursing sore fingers and his defence with a corner to defend.
The Valiants eventually breached Lincoln's rearguard but they needed a moment of magic from Whitaker to do it.
The fleet-footed flank-man twisted his escort into the ground before dispatching a rasping drive into Marriott's bottom right-hand corner from a seemingly impossible angle.
Sodje, however, was not displaying the same eye for goal as his colleague and the striker needed another two attempts to eventually find the target.
The best of the gilt-edged double landed at the striker's feet when Nat Brown made a mess of a routine clearance but, although the net was spared, seat 54 in row Z wasn't so lucky.
Two minutes later Sodje collected Colin Miles' arrowed ball before skipping past the flapping visiting keeper and slotting home.
The two-goal first-half was relegated to a mere support show as the second 45 produced an attacking Port Vale master-class which put their second-placed League Two opposition to shame.
The Valiants produced a breathtaking opening 20 minutes and were unlucky not to add to their advantage.
Whitaker combined with Leon Constantine on the edge of the box before screwing over when really he should have tested Marriott's reflexes.
Two minutes later, however, he did. The winger navigated a route in thanks to some flamboyant approach play from Sodje and Constantine and stung the palms of the keeper with a powerful shot.
Lincoln were triggered into action when Frecklington nodded home Lee Beevers' speared centre with ten minutes left to put the home side firmly on the back foot.
But despite the late flurry, Vale hung on to secure a much needed bounty and, following an exhausting four days of cup action, the fatigue stricken squad deserve a few days off.