Callum McFadzean scored his first goal for the club early in the second half at Bramall Lane to give the Blades the advantage.
The United boss told Sky Sports News: "We're happy because we did set out to win the game. We knew that was going to be important in front of our own supporters and we've done that.
"In the bigger context of it, it's half time and we know that. We've done half a job tonight and we go down to Yeovil full of confidence that we can finish the job off."
The sides meet again at Huish Park on Monday, with the winner facing either Swindon Town or Brentford in the Wembley final later this month.