Liverpool controlled much of the ball in the first half but failed to get a single shot on target during the opening 45 minutes.
Chris Wilder's Sheffield United side were impressive and limited Liverpool's play in midfield, stifling their attack and pushing forward when they had a chance.
The Dutchman sent his volley straight at Dean Henderson, however, the former Manchester United goalkeeper fumbled and let the ball cross the line to give the Reds all three points.
It was an uncharacteristic performance from the Reds and Klopp admits he aired his frustrations with the players at half-time.
The German said: "For us it was important to win, we had unbelievably big chances, in the first half we had two big chances and in the second one which we scored from.
“These games, they are not all beauties and you have to work hard for the results.
"You have to respect the opponent, we had good moments and it was one of these games you have to win but not think too much about it.
“One team deserved to win, which was us, but all respect to Sheffield United, they were tough.
"I was frustrated, and at half time we spoke about it. We started well, but you have to do the same things again and again.
“It’s about body language, it starts with the centre halves, it’s the little things.
“They threw everything at us, they worked incredibly hard, they were a real threat. I am fine with the game, not overly happy but it happens.
“We were concentrated at set pieces for 95 minutes the boys were spot on. We didn’t play our best football but there we are."
Liverpool's victory maintains their flawless start to their Premier League campaign, with this their seventh Premier League win in a row.