After a drab 90 minutes ended goalless, the Imps goalkeeper saved spot-kicks from Louis Marsh and Benie Traore to send his side through.
This was not the performance or result that the Blades will have been looking for having lost their opening three Premier League games of the season and boss Paul Heckingbottom will have been presented with more questions than answers.
Mark Kennedy’s side will now dream of drawing one of the big boys in the next round after a deserved win.
It was no surprise that United did not fire as they fell in line with the majority of Premier League clubs by making wholesale changes to their side for the visit of the Imps.
And that had an inevitable effect on the game as the Blades were unable to play with any fluency and Lincoln more than held their own.
Reeco Hackett had a curling shot saved by Adam Davies, while the visitors also wasted several good moments on the counter-attack.
The Blades had to wait until the 51st minute to have a shot on target as Ollie Norwood sent debutant Cameron Archer through on goal, but the striker’s aim was off and he shot straight at Jensen.
Lincoln finally cut the Blades open on a transition after Daniel Mandroui’s incisive pass was perfect for Teddy Bishop, but he could only stab wide from close range.
United looked the likeliest to force a late goal which would avoid the shootout lottery, but John Egan flashed a header wide and Auston Trusty could only plant his free header straight at Jensen.
That meant it would be decided from the spot and after Jensen had kept out poor Marsh and Traore efforts, the job was done when Egan skied the final penalty of the night.
Sheffield United: Adam Davies, Basham, Egan, Trusty, Bogle (Seriki 63), Coulibaly, Norwood, Brooks (Marsh 46), Larouci (Buyabu 63), McBurnie (Traore 46), Archer (Osula 63). Unused substitutes: Ahmedhodzic, Foderingham, Ben Slimane.
Lincoln City: Jensen, Eyoma, Mitchell, Roughan (Jackson 66), Burroughs, Smith (Hamilton 81), Erhahon, Brown (Sorensen 67), Hackett-Fairchild (Makama 80), House (Bishop 46), Mandroiu. Unused substitutes: Wright, Duffy, Gallagher.
Referee: P Tierney (Lancashire).