The Toffees have won five and drawn one of their last six Premier League matches, with their last defeat a 2-0 reverse at Stoke on March 4.
Sunday’s 3-0 victory over Manchester United at Goodison Park saw them creep back into the top 10 and games remain against teams below them in the table in the form of Aston Villa, West Ham and Sunderland before the final-day visit of Tottenham.
Last season Everton won eight of their last 11 matches on the way to a fifth-placed finish and Barry sees no reason why their current run should not continue.
“It is a mini-target for us to try to stay unbeaten until the end of the season,” he said.
“We are on a good run at the moment so being around the training ground it is a much better place to be.
“You can never put your finger on one aspect but it is just the way it goes, if you are winning games everyone seems to be playing better.
“These sort of things come when you are winning games, if you are losing games you are not playing well so it is always much easier when you are in a team winning games.”
On Saturday, Everton head to Barry’s former club Aston Villa, who despite reaching the FA Cup final still have to safeguard their top-flight future.
Their fortunes have also recently turned around and Barry believes that is due to new manager Tim Sherwood.
“I think he deserves a lot of credit. There was lot of pressure on him when he went in with the position they were in,” Barry told evertontv.
“The way they performed at Wembley (beating Liverpool in the FA Cup semi-final) was superb.
“But Tim knows Saturday will be a massive game for them because they are not safe at the minute either.
“It’s such a big game for them and the pressure’s off us but we still think we’ve got a lot to prove this season.”