Baines' penalty was sandwiched in between goals from Kevin Mirallas and Leon Osman as Everton won 3-1 at Goodison Park to set up a quarter-final against Baines' former club Wigan.
And the full-back told ITV1: "We've been close a couple of times and that whets your appetite. We want to go further but we've got a tough tie next.
"We're at home so we'll be favourites, but we've had tough games against them the last couple of seasons and they're coming off the back of a good win at the weekend so they'll be confident."
Oldham appeared unlucky not to have a penalty of their own when Phil Jagielka's clearance struck Darron Gibson on the hand.
And Baines added: "I haven't seen it but (if so) we'll take our bit of luck, you need it if you're going to get through in the cup."
Osman said: "We knew it was going to be tough, they're very physical and good in the final third."
And he hailed fellow goalscorer Mirallas, saying: "He's been in a good bit of form over the season and he popped up with an important goal tonight to get the ball rolling."
Substitute Matt Smith scored Oldham's consolation goal, the giant striker's fourth goal in the last two rounds of the cup against Liverpool and Everton.
Reflecting on the result, he said: "It's disappointing, it's been a good FA Cup run for the club.
"We played very well and we're not going to go up against a stronger opposition all season so we can take some credit from the performance."
Smith was left on the bench for both games against the Toffees and said: "You want to start every game, especially as a forward to get a run and breed confidence in front of goal.
"But I'm not one to my throw toys out of the pram, I just want to help the team once I get on."