Christian Eriksen's late leveller saw Tottenham avoid extra time and record a 3-2 aggregate win over the third-tier outfit at Bramall Lane.
Tottenham, who won the first leg 1-0, seemed headed for a Wembley showpiece with Chelsea, after a glorious 28th-minute free-kick from Eriksen.
A quick-fire double from the home side's teenage substitute Che Adams turned the match on its head though, and extra-time seemed likely - until Eriksen struck again.
"For sure I was worried - it is football!" Pochettino told the BBC.
"Over the two legs we deserved to go to Wembley but you need to create chances, score and kill the game.
"We gave the possibility to Sheffield and when they scored the second goal, it was difficult. Football is emotion.
"Now we are happy for the players, they deserve the right to play at Wembley and for our supporters it is the most important goal.
"It is my first final at Wembley, it is new to me and new to a lot of our players, but we will enjoy it."
Tottenham striker Harry Kane was elated with the result and has dubbed the meeting with Chelsea as a "decider" for the season between the two clubs.
Chelsea beat Spurs 3-0 in December but earlier this month, Kane struck twice in a memorable 5-3 win at White Hart Lane.
Kane told Sky Sports: "It means the world [to make the final].
"We were in cruise control and that was the worst thing we could have done. Credit to Sheffield United but it shows how much character we have.
"At Wembley anything can happen.
"It's 1-1 between us [and Chelsea] this season, it's a decider if you like and there's no better place for it than Wembley."
