The Toffees have picked up back-to-back wins over Manchester City and Stoke City and are just four points behind Chelsea in fourth place.
Howard admits Everton will need a bit of luck but insists they intend to push their rivals to the finish.
"We are right there, we are breathing heavy down a few people's necks," said the United States international.
"We need a little bit of luck for those around us to drop points. But maybe we are in a position to make our own luck. People say it's daunting but it's an opportunity for us."
Howard returned to the side for the victory over the Potters following a back injury, while Phil Jagielka had a pain-killing injection to play and the keeper feels this shows the desire of the team.
"That is just the trust level and passion we have," added Howard.
"At other clubs you might get certain guys saying 'I will take another week and then come back'. But for me and Jags that never came into it - we love to pull on the shirt."