Prior to the victory at Goodison Park, the Toffees had failed to win in four outings but Round feels that perhaps that winning formula was again evident despite beating a League One opponent.
"I thought there were signs tonight that our flow was coming back," he told the club's official website. "Recently we had lost a little bit of that.
"I know we were against opposition lower than the Premier League but there were signs that our cohesion was coming back. The way that Steven Pienaar and Leighton Baines, and Kevin Mirallas and Seamus Coleman were linking up - we showed that maybe we are starting to return to some sort of form.
"It was a tough game I thought. Oldham put a right shift in and pushed us all the way. I thought they were a credit to the manager and themselves. They acquitted themselves right and made it very tough.
"Having said that, we matched them with our attitude and the quality came through in the end. You saw that with one or two of the goals that we scored. Unfortunately we missed a couple of chances to make it that little bit more clinical."