Ellington had scored three times against Derby when Bristol Rovers pulled off an FA Cup shock in January 2002, but on this occasion the striker was the saviour for Paul Jewell's side.
Lincoln had taken a 48th minute lead through Ben Wright, but Ellington equalised seven minutes before the end and then struck twice in extra-time.
The result was hard on Lincoln who had battled bravely and might have won the first round tie but for a great save by Roy Carroll that saved Derby from going two goals behind.
But the Championship team had the chances to have taken control before half-time when Ellington's finishing was not as sharp as it was later in the match.
He was through on goal in the seventh minute but could not beat goalkeeper Rob Burch, who also did well to keep out a 20-yard shot from winger Steve Davies.
Robbie Savage should have done better when a mistake by Lee Beevers put him through, but his control let him down and those misses threatened to prove costly when Lincoln took the lead.
Paul Green whipped in an excellent cross from the right touchline and Wright glanced a header past Carroll from six yards.
It was an excellent goal and Derby were clearly shocked and could have been in deeper trouble in the 70th minute.
Dany N'Guessan broke away down the Derby right and picked out Kevin Gall, but Carroll dived to his left to block the striker's close-range shot.
The home fans were becoming increasingly desperate when the visiting defence was opened up by a deep cross that Ellington squeezed inside Burch's left-hand post.
Derby almost scored again in stoppage time but they dominated the extra 30 minutes with Ellington finishing clinically from 12 yards after a strong run from Kris Commons.
Five minutes later Ellington finished Lincoln off when a headed clearance fell invitingly for him and his right-foot volley left Burch a spectator to book Derby's place in the second round.