The Blues went into the match as heavy favourites but made hard work of the spirited Blades, who came agonisingly close to finding the back of the net on a number of occasions.
David McGoldrick spurned a golden first-half chance to head home from point blank range, while former Liverpool striker Rhian Brewster narrowly missed the target with a late strike from the edge of the box.
Despite Sunday’s match marking Chelsea’s seventh clean sheet in a row in all competitions, Tuchel was less than impressed with his side’s efforts after the final whistle.
The German boss told BBC One that the Blues had luck on their side, claiming that his players were fatigued after avoiding defeat in each of their last 14 matches.
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“We were clearly tired now. I could see after 14 matches consecutive. I could feel us tired, a lack of concentration, many little faults and strong opponent in the second half.
“We were lucky in the second half to keep the clean sheet. It was not the first time in 14 matches when we were lucky and allowed too many chances, but it can arrive and it’s important in the end to go through.”
Sheffield United produced a much stronger performance than the result gave them credit for, with Jason Tindall’s side looking far better than last week when they were subjected to a 5-0 hammering at the hands of Leicester City.
“We let them come back, because like I said we had full control in first half and a big chance with Christian [Pulisic] to finish the game early,” added Tuchel.
“After that we had too many mistakes, easy mistakes. We lost duels and we lost a bit of concentration and momentum.
“Honestly I could feel it in training, since some days we feel a bit tired and have a lack of concentration, it’s not too easy. It’s normal. It can happen.
“I’m happy we could help from the bench with the guys who came in. In the end we have a win and take it.”
Chelsea will be looking to continue their winning ways with another victory over Premier League strugglers West Bromwich Albion on Saturday, ahead of a return to Champions League action four days later.
The Blues will take on Porto over two legs for a place in the competition’s last four as Tuchel seeks European glory at the first time of asking.