A 2-0 defeat at Southampton on Saturday has put a serious dent in Everton's UEFA Champions League ambitions, with an untimely setback leaving them in fifth spot.
They are now a point behind Arsenal having played a game more and have just two more fixtures in which to try and force their way onto European football's top table.
Howard believes they can still hit that target, with a positive response expected from Roberto Martinez's side after own goals from Antolin Alcaraz and Seamus Coleman saw them come unstuck at St Mary's.
The American goalkeeper said when asked if Everton are now out of the race for fourth place: "Absolutely not. We still have a chance mathematically so therefore it is not the end.
"We need to win and play better. There are still six points remaining, if we win and Arsenal drop points we are in the Champions League.
"You can't legislate for own goals, especially two in the space of 30 minutes, but it just happens and you deal with it and move on.
"We are trying with every bone in our body but that is part of being at the top, part of the pressure that professional athletes thrive on, and it has not been a problem.
"We are a top team and dropped points but Arsenal can drop points as well. We have been putting in brilliant performances all season so will expect a good display next week (against Manchester City)."