Garcia replaced Gus Poyet at the Amex Stadium over the summer and currently has the team in eighth place going into a home game against Huddersfield this weekend.
Brighton finished fourth in the Championship last season, suffering a play-off semi-final defeat to eventual winners Crystal Palace, and Barnes says getting back into the top six this season is the Seagulls' minimum expectation.
He told Sky Sports News Radio's Football League Hour: "It was quite hard to take, missing the chance to go to Wembley which would have given us the chance to get into the Premier League.
"But a few new faces have come into the dressing room since then and we're building to try and push again this season.
"Our aim, like every season, is to be in the top two but if you can't make top two then we have to make the play-offs and our main aim is to be sure we're in that top six come the end of the season.
"It's going brilliantly. We had a little blip against Barnsley a couple of weeks ago but since then everything has been positive.
"I think we've gone a little bit more direct this year as last year it was all about passing on the floor and playing out from the back.
"We're still playing football from the back as and when we can but we're definitely more direct."
It was one of the lesser lauded elements of the game, a corner, that Brighton created their winner at Middlesbrough last weekend, and Barnes admits set-pieces were something Garcia had focused on ahead of the game.
He said: "We were on the back foot. We started off well in the first 10 minutes but from then we defended for our lives. They were amazing in attack with (Mustapha) Carayol and (Lukas) Jutkiewicz but luckily our defence held strong and then we won it with a set piece late on.
"We knew they had conceded a lot of goals from set plays and that was our target (to capitalise on that), and we come away with a win with a set play."
Barnes expects another difficult game against Huddersfield but hopes he can continue his recent good form in front of goal to help his side to a win which could lift them into a play-off position.
He added: "Huddersfield are a brilliant side but every team in this league has a brilliant side and are difficult to beat and we go into every game wanting to win.
"I'm not as happy as I could be (with my goal tally this season). From a striker's point of view you always want to score as many goals as possible but lately I've picked up form and scored five in my last six games so long may it continue."
Also on this week's show are former Barnsley manager John Hendrie, Derby captain Richard Keogh, QPR striker Charlie Austin, Rotherham captain Craig Morgan, Northampton goalkeeper Matt Duke and Cheltenham boss Mark Yates.
Listen to the full show via the Soundcloud player below.