Baines scored from the penalty spot as David Moyes' side beat League One Oldham Athletic 3-1 in a fifth round replay at Goodison Park on Tuesday evening.
The win sets up a quarter-final tie at home to Wigan Athletic, with the Merseysiders eyeing a place in the semi-final for the third time in five seasons.
Everton lost to city rivals Liverpool in last year's semi-final at Wembley and although they went one better with victory over Manchester United in 2009, they then went on to lose the final to Chelsea.
Baines believes that previous cup heartache is spurring the squad on and making them determined to go all the way in the competition this year.
"We have gone close to winning it a couple of times and so it's whetted the appetite," said the England defender.
"We feel we have unfinished business with the FA Cup. We want to go that bit further and win something now."
Looking ahead to the next round, Baines had a word of warning for his team-mates, adding: "We are on the right track but we have another difficult game in the next round against my old club, Wigan.
"We will be favourites at home but we have had tough games against them over the past few years.
"They have just come off the back of a good win so their confidence will be up."
Fellow full-back Seamus Coleman believes the Oldham result, which ended a run of four games without a win, can kick-start Everton's season.
"We were disappointed at the weekend [defeat] against Norwich," he told the Liverpool Echo. "I thought we played some good football at times but didn't see the game through, and after a result like that you want a game as quickly as possible and thankfully we had one.
"It's a win and we can go into the game against Reading [on Saturday] a bit more confident now.
"We know Wigan will be hard but being in front of our home fans is always going to give us an advantage.
"We would like to get back to Wembley but every team is looking for the same thing, so we can't think too far ahead.
"Wigan got a good result last week and we just hope we can do a professional job and get there at the end of the season."