The result sees Chelsea remain unbeaten in 14 matches in all competitions since Thomas Tuchel took charge of the club.
Chelsea have also kept a clean sheet in their last six matches and are yet to concede at home under Tuchel.
"Sheffield United didn't make it easy on us -- they have quality and we had to suffer a bit in the second half. But it's another clean sheet, a good performance," Pulisic told BBC.
"We have a confident bunch of guys here and when you go on a streak like this we're feeling really good and we're feeling like we can beat anyone."
However, Chelsea grabbed the opening goal after 24 minutes when Ben Chilwell gathered the ball from a corner and his strike was turned into his own net by Norwood.
Pulisic almost doubled Chelsea's lead before the break after capitalising on another Norwood mistake but his effort was blocked by goalkeeper Aaron Ramsdale.
The American went close to scoring minutes into the second half but Ramsdale denied him again with a good save.
The visitors should have equalised after 67 minutes but an unmarked McGoldrick missed a header from close range.
"It sounds excellent to say we're through to the semis - it was a tough match. We had a good first half where we controlled everything," Tuchel told the BBC.
"We lost control in the second half, we were clearly tired. After 14 consecutive matches I could feel we were tired and we made many little faults.
"We were lucky to keep the clean sheet in the second half. It's important to go through in the cup. That was the target and we reached it."