Leading 1-0 from the first-leg, Spurs appeared to be coasting towards a final with Chelsea after Eriksen???s stunning free-kick on 28 minutes appeared to have put the tie beyond the reach of Nigel Clough???s League One side.
The Blades struggled to create any real chances of note but, with 15 minutes to play, two goals in two minutes from substitute Che Adams turned the tie on its head and looked to have sent the game into extra time.
Fellow substitute Louis Reed looked had a glorious chance to send the Blades through but it was to be Eriksen who would have the last word as he guided Spurs??? second of the evening past Mark Howard with two minutes remaining.
Spurs edged the tie 3-2 on aggregate and now head across London to play Chelsea in the final on March 1 ??? a repeat of the 2008 League Cup final.
Having rested a number of first-team regulars in Saturday???s FA Cup defeat to Leicester, Mauricio Pochettino reinstated Harry Kane to the starting line-up and, after battling the conditions and a cacophony of noise at Bramall Lane, Spurs soon got into their rhythm.
Michel Vorm, who retained his place after his FA Cup error, almost gifted the hosts the lead on nine minutes as he spilt Campbell-Ryce???s cross at the feet of Marc McNulty, who set up Jamie Murphy, but his effort was locked on the line by Kyle Walker.
Kane was lively from the off, harrying Jay McEveley in the opening exchanges, and hitting the post after Walker caused problems down the right against his old club.
One of many marauding runs from Ryan Mason drew a foul 25 yards from goal, presenting Eriksen with a chance to open the scoring, and the Danish international duly obliged with an exquisite curling effort which left Howard rooted to his spot before going in off the far left-hand post.
Going behind stunned the Blades and their home support and Spurs, without threatening to extend their lead, controlled the remainder of the first half in the Steel City.
Mousa Dembele ensured Spurs started the second half as they had finished the third, taking the game to the hosts which a mazy run inside the opening minute.
Spurs looked on track for their first Wembley final in eight year until Clough introduced Adams with a little under 20 minute remaining, and within five minute the match was level.
Ryan Flynn delivered a dangerous cross along the six yard line from the right and, after the ball evaded walker at the far post, Adams shot first time under the reach of Michel Vorm to give the hosts hope.
And his deflected effort less than 90 seconds later sent Bramall Lane into raptures as Spurs passage into the final hit the rocks.
Reed could only guide a fiercely-struck effort over the bar from 20 yards after he was found in space on the right-hand edge of the box and Eriksen???s punished the Blades four minutes later.
Kane was the provider, picking out the midfielder who, after evading the united defence, slipped the ball into the bottom right-hand corner to seal the North Londoners passage to Wembley.