The Dane had already curled home a stunning free-kick when he struck again two minutes from time to rewrite the headlines after teenager Che Adams had climbed off the bench to net a quick-fire double and turn a thrilling match on its head.
Eriksen's glorious 28th-minute set-piece look set to send Tottenham on their way to a re-run of the 2008 League Cup final, when the White Hart Lane outfit were 2-1 winners - their last trophy success.
But United boast a stunning record in knockout competitions under Nigel Clough, with their first-leg reverse - inflicted upon the League One side by Andros Townsend's second-half penalty - only the third defeat they had suffered in 21 previous cup games under his guidance.
And Adams - who only joined from non-League Ilkeston in November - was in no mood to allow that tally to be added to, the 18-year-old coming on for the final quarter of an hour and scoring his first two professional goals in an incredible three-minute spell.
But, after another United substitute, Louis Reed, had blasted a shot just over from inside the box, Eriksen provided the killer blow to end an incredible night at Bramall Lane.
Last season's FA Cup semi-finalists took to the task of overturning their one-goal first-leg deficit with great zeal, setting a tremendous early tempo that Tottenham were more than happy to match.
United spurned a great chance to level the tie in the 11th minute when Jamal Campbell-Ryce's teasing cross was spilled by Michel Vorm under pressure from Mark McNulty, but the Dutchman was able to deny Jamie Murphy on the rebound.
Tottenham, still smarting from the FA Cup loss to Leicester City at the weekend, highlighted their threat when top scorer Harry Kane linked with Erik Lamela before firing a powerful effort straight at a grateful Mark Howard.
But there was nothing Howard could do about the opening goal as Eriksen struck with unerring precision from 20 yards, his free-kick hitting the inside of the left-hand upright as it dipped just under the crossbar.
The tie could have been put beyond United's reach just two minutes into the second half, but Kane dragged wide from close range to grant Clough's men a reprieve.
It was a let-off that preceded a prolonged spell of controlled possession from United, though there was a sense that Tottenham were content to sit off and contain their opponents.
That proved to be a huge error in judgement as Adams struck twice in quick succession to turn the tie on its head.
First, he turned home Ryan Flynn's low cross at the back post in the 77th minute and then, with Tottenham rocking, he provided the finish to Murphy's cross to send Bramall Lane wild with 11 minutes left.
The visiting fans, who had been revelling at the prospect of a trip to the national stadium, were now stunned into silence and could only watch on as Reed blasted over from a tight angle with six minutes remaining.
Just as it looked as though the momentum was with the hosts, though, Eriksen latched on to Kane's pass and found the bottom corner to win it for Tottenham.
There was still time for Chris Basham to clash with Vorm when going for a late throughball, but Tottenham ultimately secured a Wembley meeting with Chelsea on March 1 - just three days after Mauricio Pochettino's side are scheduled to face Fiorentina in the UEFA Europa League.
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