The Toffees are now two points behind sixth-placed West Ham United with three games remaining in the season.
Asked whether he believed Everton's display was worthy of three points, Pickford said: "I think in the second half we were better and we created a few chances.
"The Villa keeper had some good saves and they made a few good blocks. I think in the second half, definitely [we did enough].
"In the first half, we didn't quite get going – but we were still in the game. And we gave ourselves the best chance of getting a result by keeping a clean sheet and having a solid base at the back.
"I think we grew in belief in the second half. We started to find those gaps – Gylfi, Richarlison and Dominic started getting chances. We just needed one of our chances to go in.
"We'd like to have won, of course, but we take a point and that could be crucial at the end of the season.
"We've got three big games now and it starts against Sheffield United on Sunday – and we'll go for the win."