Noel Hunt gave Reading a fifth-minute lead, but the League One Blades came in to the game in the following half-hour only to see Mikele Leigertwood give the home side a two-goal cushion with a fine 25-yard drive against his former club.
Hunt's second, and a strike from Garath McCleary, put the result beyond doubt in the second half. With rivals Norwich and QPR both being knocked out by lower-league opposition, McDermott was happy his side got themselves in the hat for the last 16.
"If you look at the results today, a couple of big teams have gone out and we are thankful we are not one of them," the Reading manager said.
"I don't think we did particularly well in the first half. We just weren't ourselves. I got a bit agitated, which is unusual for me.
"But we were much more clinical and efficient after, so we deserved to go through."
Having looked like certainties for relegation towards the end of 2012, Reading now find themselves welcoming Chelsea to the Madejski Stadium on Wednesday on the back of a four-match winning streak.
The high morale in the camp bodes well for the Berkshire club's chances of avoiding the drop and also boosts their hopes of maintaining their strong history in the FA Cup, where they have reached the quarter-finals in two of the last three years.
"The confidence has always been good here, but winning games always breeds momentum," the 51-year-old said.
Given the poorer quality of opposition, McDermott can also take heart from the fact that the likes of Hunt, Nicky Shorey and Adam Le Fondre got 90 minutes of football to stave off any frustration they may have had at not starting regularly this term.
"I tried to keep them all on the pitch for 90 minutes because a lot of them haven't played full games so they needed the minutes under their belts," the Royals boss added.
"It was nice that Hunty got a couple of goals under his belt and Garath has made one and scored one today so it's fantastic for him."