A Kelvin Lucas double gave the hosts a half-time lead before Alex Webber pulled a goal back for the Mavericks ten minutes after the restart. With minutes left on the clock, sub Joe Warner sealed Storrington's first win in five games.
Everett said: "I felt it was a must win game for us so the win was really pleasing. We had some good moments in the game but it wasn't straight forward.
"I actually think it would be fair to say that United created the more clear cut chances so we have to be thankful that we were so clinical and in particular, Kelvin, for putting us two up. We did things better than them in the penalty boxes today and that was the difference.
"The win takes us closer to the top eight in the league but a defeat would have seen us looking over our shoulders so that is why I put such great importance in the win."
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The Mavericks hit back on 56 minutes as Webber's header at the far post beat Gary Elliott in the Storrington goal and this lead to a spell of serious pressure from the visitors as the Swans seemed to tire.
Everett's side did, however, get a third against the run of play as sub Joe Warner netted the third on 90 minutes to take the Swans up to tenth.
Storrington travel to second-place AFC Varndeanians on Saturday and Everett feels that his team need to start taking points off opponents that are higher up the table.
He added: "This coming Saturday will be tough looking at the results Vardeanians have produced. But we've seen some of our best work against the better sides.
"We need to start taking points off the teams above us, that's something we haven't done enough of. If we play with the focus and desire I've seen in the last seven days there's no reason we can't get a positive result."
Storrington: Elliott, Hide, Shoebridge, Setchell, Jarvie, Stideford, Gilmour, Cave, Clarke, Lucas, Suter. Subs: Joe Warner, Dean, Lamb.