Everitt kicked 11 of his side's 16 points while his opposite number Shane Drahm missed a penalty in nearly the last kick of the game which would have given the league's bottom club a morale-boosting win.
Smith said: "Barry took every kick that was available to him with the exception of the one that was out of his range.
"It was a good contest between Drahm and Barry. I have seen Drahmy bury a lot of teams with the last kick of the game and he went close to doing it again.
"It was a good arm wrestle with plenty of drama but it won't go down as an epic. I thought we got off to a great start but sometimes in sport when you get off to a flying start like we did - we were 10-zip early on - you relax."
Everitt's penalty put the visitors ahead in the third minute before his kick was picked up by Topsy Ojo to cross after 10 minutes.
Two penalties and a drop goal by Drahm brought Worcester to within one point but Everitt responded with two penalties of his own.
The real drama unfolded in the final 10 minutes when Aisea Havili grabbed a try for Worcester.
With an opportunity to level the scores Drahm's conversion hit the post and then he proceeded to miss a penalty to win it with time almost up.
Worcester's director of rugby John Brain tried to take the positives from the defeat.
He said: "We've lost, which is disappointing, but the fact that we came so close means we can take something out of it.
"They are a potent, attacking team but we soaked up a lot of it, particularly in the second half.
"We could have nicked the game but they probably just deserved to win. They came with a game plan and stuck to it.
"Although it is a blow to our momentum to lose, we haven't conceded four or five tries.
"It was a war of attrition but the next game against Harlequins will be a different game - they have a different way of playing."