Despite going into the break goalless, a second-half brace from Chris O'Grady and goals from Liam Lawrence and Dale Jennings saw Barnsley leapfrog their Somerset hosts and secure back-to-back wins for the first time this season.
Hutchings, who was appointed as assistant manager to Danny Wilson on Sunday, said: "The boys have been working very hard off the back of a midweek game and a very good result at Reading, and today we took our chances, which is what it's all about.
"It was an even first half but the boys are playing with confidence and they have to take that into the remaining games.
"If you're going to be successful you need a goal scorer and Chris [O'Grady] is doing that, but we had 11 heroes there and we need to take that forward.
"All we can do is take care of our own results and make sure we go into every game with the same positive attitude and come out with three points.
"I think you've seen today and on Tuesday that we're up for this battle.
"We've worked hard in training and implemented one or two things we expect of the players and they've carried it out.
"A big 'thank you' goes to the fans today as we had a fantastic turn-out. It's a long way to travel, especially with the position we're in, and we hope they have a good journey home."
Yeovil manager Gary Johnson accused his side of being "too nice" after their winless run extended to six matches.
He said: "It was a poor performance.
"Barnsley turned up and looked stronger, quicker and cleverer in every department, and despite going in at half-time with a strong wind against us, we felt if we were a bit better with our control of the ball, our passing and our runs we could get something.
"Their first goal was a poor one and our boys didn't seem to put any part of Yeovil's game together. It was like a group of lads off the street that we put together, threw them out there and asked them to play.
"It's very disappointing and it's probably our worst all-round performance.
"Every game you lose it looks more difficult, but until it's mathematically not possible we'll keep going, but if we play like that then we'll be bottom by a million miles."