A 3-0 victory over Arsenal on Sunday moved the Merseysiders into fifth, one point behind Arsene Wenger's men, having played a game less than the north London side.
Everton have also recorded victories over Manchester United and Chelsea this season and Baines thinks those results will give the squad added belief they can finish in the top four.
"We are perfectly comfortable being underdogs or having the expectation - it is probably nice to have that little bit of expectation to drive us on," Baines said.
"We are aware of the task in hand and we have some tough games to play but they are here at Goodison where our form has been exceptional and we will have confidence going into those games.
"We have done well against a lot of the big sides. We have done well against Arsenal in the league, done well at United and played well at Goodison against Chelsea and Tottenham so we have held our own in that group against what you consider the top sides.
"We have a few games left to try to push on and the fact we are in this position is great for us."
The England defender believes Everton will be able to attract quality players should they qualify for the Champions League but insists the squad are taking things one game at a time.
"We have a game in hand (at home to Crystal Palace) coming up and until we get to that point where we have played the same amount of games and have worked ourselves above Arsenal then it probably still feels like we are playing catch-up," he said.
"It is hard because you don't want to get ahead of yourself and start thinking about the Champions League too much but we all know what it means to any club, particularly one who hasn't been competing in those circles for a long time.
"It would mean great things but we don't want to get too far ahead of ourselves at this point.
"You look at the top four and it changes. Liverpool have worked themselves into that bracket and United are not quite there at the moment but they will come back stronger.
"We will try to work our way in among that pack if we can but it has always been difficult because we haven't always had the funds to go and spend but if we could get into a competition like the Champions League then it would help in that aspect.
"It raises the bar a little bit and makes the club more appealing to potential signings and gives the manager, the chairman and the board a little more scope financially to push themselves in the market more.
"There are a lot of things that qualifying for a competition like that can do but there is still a lot of work to be done for us to get there."