Tottenham, who will face Chelsea in the final at Wembley on March 1, appeared to be cruising when they took a first-half lead through Eriksen's stunning free kick which took their aggregate advantage to 2-0.
Sheffield United teenager Che Adams came off the bench, however, to turn the tie on its head with two goals in two minutes, giving his team a 2-1 lead and levelling the aggregate score.
Eriksen ensured four-times winners Spurs avoided playing extra time when he found space on the edge of the lower league side's box with two minutes remaining and calmly slotted home.
"Sheffield United definitely gave us a big fight. They gained confidence after they scored their first goal, the second one was a bit lucky but we came back at the end," Eriksen told Sky Sports.
Eriksen's opener on 28 minutes was a moment of supreme quality.
The Dane whipped a stunning free kick from the right side of the area into the far top corner off the post with the hosts' keeper Mark Howard rooted to the spot.
The lower league side, whose cup pedigree has upset five Premier League sides in the last 12 months, laboured to claw a way back into the match, but the more they pushed on the likelier it looked that Spurs would add to their tally.
Yet two minutes of madness turned the tie around and left Tottenham facing a scrap they would not have seen coming.
Adams came off the bench on 74 minutes and kept his head to finish calmly past Vorm with his instep, sparking the home crowd into life and adding spice to a cup tie that had seemed to be heading to a predictable conclusion.
His second goal had an element of good fortune as his deflected effort from the edge of the area looped away from Spurs keeper Vorm.
With their tails up, United looked favourites to grab the winner, but their hopes of a reaching a first final were dashed by Eriksen's precise finish.
"It was just a bit of quality that made the difference," Sheffield United manager Nigel Clough said. "Eriksen's two goals tonight were superb."
(Editing by Ed Osmond)
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