A second-half Andros Townsend penalty was all that separated Spurs and their League One visitors, with the tie very much still alive heading into next Wednesday's second leg at Bramall Lane.
Including the spot-kick, awarded for a handball against Blades defender Jay McEveley, Pochettino only saw his side hit the target twice on a night when Sheffield United frustrated their top-flight hosts for much of the contest.
And the Argentinian said he had to change his side's approach at the interval as they had struggled to penetrate a resolute defensive unit.
"I think it was a very tough game," Pochettino said.
"If Sheffield United are in the semi-finals, it is because they deserve (to be there)
It was difficult to create chances
They played deep, had a good organisation, played aggressively and maybe we were wrong in the way we tried to arrive and create chances.
"I think we improved our performance in the second half and created chances
I think we deserved our victory.
"Maybe our position was wrong, I think we were wrong in our defensive position, we played very slow and had a low tempo
(There were) always three, four, five touches and after that we passed the ball
This made it difficult to create chances.
"Yes, we try to move and change the position, we tried to create problems to the back four
It was a team problem and not a one-player problem
In the second half I think we fixed things."
Adebayor, making his first start in almost three months, also led the team out in the absence of the rested Hugo Lloris and Younes Kaboul.
The Togo striker missed a great chance in the first half after picking up a booking for catching Blades midfielder Louis Reed in the face and was also jeered by some of the home fans when replaced by Roberto Soldado.
But Pochettino made clear the decision to install Adebayor as captain was made in the summer and he had no qualms in giving him the armband here, despite his tempestuous relationship with the Spurs support.
"He is one of the three captains," Pochettino said.
"You forget (I named them) at the beginning of the season
I think it is not the moment to explain why
When I took a decision it was seven months ago, I took the decision that Kaboul, Lloris and Adebayor were three captains, seven months ago
It is a long, long time ago that happened.
"I had no doubts (about making him captain)
I didn't hear (any bad reaction)
I am focused on the game."
United boss Nigel Clough was left agonising over the penalty decision having seen his side push their more illustrious opponents all of the way - although the scoreline was one he revealed he would have settled for ahead of the game.
"We made it as uncomfortable for Spurs as we could," he said.
"They found it difficult to break us down
We could have created more but I was pleased overall
We would have taken the result if you'd offered us it before the game
But we're disappointed because we played well."
"We are still in the game
You heard the noise 4,000 fans made here and there will be 24,000 of them next week.
"Jay McEveley was appealing for offside
He is devastated
He said the ball just checked back and hit him on the arm."
Kick-off was delayed for 15 minutes as the United team bus was stuck in traffic, but Clough was not ready to use their travel troubles as an excuse.
"It took us longer to do the five miles than it took us to get to London from Sheffield," he said.
"But it didn't affect us
We were very well organised and hard to break down
It was not a nice way to lose a game."
Source : PA
Source: PA