Oak - beaten 6-1 at promotion-hopefuls Lingfield last time out - looked to be heading for a second straight defeat at rivals Storrington as they trailed 2-0 at the interval.
However, Shoulders’ side served up a fine response with second half strikes from Jamie McKenzie, Andrew McDowell and Liam Austin-Slade firing Oak to a 3-2 triumph.
Mile Oak’s hopes of a top-three finish were left in tatters after the defeat to Lingfield but the win leaves Oak - in fifth - five points off Lingers having played a game more.
And Shoulders wants the second half performance at Swans to be replicated for the rest of the season.
He said: “It was good to get the win after last week. Last week was very bad for us, we had really bad injury problems which only got worse in the game so we had to quickly move on.
“The game at Storrington was in awful weather and the pitch was in a right state by the end of the game.
“The lads did what I asked in the second half, putting Storrington under pressure like what they did us in the first half.
“We got the three goals, which could turn out to be massive in terms of where we finish this season.”
Swans were two goals to the good at the break. Dean Gilmore got the first on 13 minutes, then Joe Warner added a second a minute prior to the interval.
Frontman McKenzie got one back two minutes after the restart, before McDowell levelled things up less than a minute later.
Austin-Slade then completed the turnaround nine minutes from time as Mile Oak ran out 3-2 winners in the end.
Mile Oak are back in league action on Saturday as they entertain Ringmer.
MILE OAK: A.Stenning; Early, Pollard, French, Stevens; Edwards, Austin-Slade, Wedge-Bull; McKenzie, Furnell, McDowell. Subs: J.Stenning (Furnell), Wells (McDowell).