Staines made the relatively short journey to Hampshire not in particularly fine fettle themselves, having lost their last two, including a 7-0 hammering by newly-promoted Brentwood Town on Saturday.
And they started this one on the backfoot, goind behind after 33 minutes to a Bobbie Dale effort.
But 5 minutes into the second half, Luke Wanadio equalised, only for Dale to restore Boro`s lead less than 2 minutes later.
Ryman League
Eight minutes from time, substitute Jack Hutchinson equalised and then Pat Cox won it for the Swans with less than 2 minutes to go.
Follow us: @rymanleaguenews on Twitter | Ryman League group on Facebook
Keep up-to-date with our exclusive email newsletters.
Subscribe