The Toffees have kept themselves in contention for a UEFA Champions League spot with a run of only one defeat in eight.
Five victories have been collected during that stretch, meaning David Moyes' men sit sixth - just five points adrift of fourth-placed Arsenal.
Howard concedes that it will be a big ask to overhaul the Gunners, and their north London rivals Tottenham Hotspur, but the American goalkeeper is refusing to give up the fight.
He told Sky Sports News HD: "I think we've always been a dark horse, we've always been the outsider in that race, and we still are.
"If we put a few moments of magic together, if we can win out, which is never easy with the games that we have, and we can get some help in and around us, anything is possible."
The next hurdle for Howard and Everton to cross comes in the form of a derby date with Merseyside rivals Liverpool on Sunday, and the Toffees are aware that the Reds cannot be taken lightly after hammering Newcastle United 6-0 in their first outing without the suspended Luis Suarez.
Howard added: "All the talk has been about how Suarez has carried them, maybe rightfully so, but boy have they made a statement and shown everybody that they also can find goals from other places.
"They are a threat, we know that, and I think we'll have mapped out a plan to try and beat them come Sunday."