City took an early lead but then survived a second-half battering as Dagenham did everything but score.
Lincoln then gave the scoreline a flattering look with a goal deep into stoppage time as the Daggers were caught on the break.
Lincoln got off to the best possible start taking the lead after 15 minutes with a well-worked corner move. The ball was played short to Paul Green and his perfect cross from the right was headed home by Ben Wright at the far post.
Lincoln kept up the pressure and five minutes later another cross from Green gave Jamie Forrester a half chance, but the striker's final shot lacked power and Daggers keeper Ed Thompson saved easily.
Lincoln had a let-off on the half hour when a poor clearance by keeper Alan Marriott went straight to Jon Nurse. The ball came back into the box, but Sam Sloma was unlucky to see his shot deflected from a corner.
Marriott blundered again a minute before the break when he completely missed a cross from Sam Saunders, but the Lincoln defence managed to scramble the ball away.
The second half was a different story with Dagenham in total control. They looked to have equalised in the 59th minute when substitute Jamie Taylor got behind the Lincoln back four, but slipped as he was about to shoot with the goal at his mercy.
Four minutes later Ben Strevens was unlucky when his low shot appeared to be heading for the net until it struck Green and rolled wide.
Dagenham kept up the pressure but Glen Southam wasted an opening when he got away down the right, producing a poor cross which eluded Ryan Hall.
Lincoln survived some difficult moments, with Lee Ridley heading a Strevens cross over his own bar.
Late on Dagenham were denied a penalty when Jon Boardman appeared to be bundled over inside the box.
Then Marriot appeared to carry the ball over his own line, but was saved when referee Clive Oliver ruled he had been fouled.
Lincoln hung on and then broke away in the fifth minute of stoppage time, with Mark Stallard setting up Dany N'Guessan, who fired into the bottom corner to seal the victory.