United fell behind after just four minutes, and never looked like getting back into the game.
They had conceded five by the end of the match, with Michael Collins' goal nothing more than a consolation.
"We spoke of the desire and passion needed before the game and at the start of the second half we showed what we can do," said Baraclough.
"Why that wasn't there from the beginning - it needed to be. You've got to get about them and when we did we showed we can create something.
"It was a great goal but a little 20-minute spell isn't good enough. It was almost a training game for them really, we were just making up the numbers, but we will take positives out of it."
Steve Round, the Everton assistant boss, hoped the result sparked an improvement in Premier League form.
"If we did start to score we knew we could go on a run, push up the league and go through in cups," he said.
"We are very pleased. I think we have been threatening that sort of a result for a while. It is just pleasing to get the goals and go through to the next round.
"It is very important to score early because the opposition can get up a head of steam if the crowd gets behind them. It did kill the game a little bit and make the opposition flat, but they came back at us in the second half.
"They came back firing and got the goal back but our quality told in the end."