Richard Hill's side twice fell into arrears at the Woodspring Stadium, but stikes from Dale Binns, Jai Reason, Damian Scannell and an own goal brought them back for 4-2 victory, pushing the Spitfires up to tenth in the Blue Square Bet South.
Eastleigh had an excellent opportunity to open the scoring when Craig McAllister pounced on a stray pass by Naby Diallo and carried the ball 20 yards before forcing a good save out of Lloyd Irish, writes James Wright.
But the visitors fell behind in the seventh minute when Ross Flitney parried a shot by Dayle Grubb only for Ross Stearn to snaffle the rebound.
The Spitfires were behind for barely four minutes. A cross from the left by Dale Binns, restored to the starting line-up in place of Luton Town loanee Alex Lacey, got caught in the gale gusting off the Bristol Channel, completely deceiving Irish and plopping inside the far post.
After such a frenetic opening it was perhaps unsurprising there was a half-hour lull.
But as the interval approached, things warmed up again with Eastleigh captain Glen Southam flashing a shot just over and visiting defenders Mark Hughes and Mitchell Nelson collecting bookings.
The second half began identically to the first with Stearn giving Weston the lead and Eastleigh immediately levelling.
The Weston midfielder converted a cross from the left at the second attempt after Flitney had blocked his first effort. But, within minutes, Damian Scannell had skinned home right-back Jamie Price to set up Jai Reason for another equaliser.
Eastleigh took the lead for the first time on 55 minutes when Scannell spun on a sixpence to squeeze a shot just inside Irish's far post.
And, as the promotion-chasing hosts pushed up in search of an equaliser, Eastleigh doubled the advantage when Binns's low centre from the left was deflected home by a defender under pressure from Reason.
With Weston resigned to defeat against a team they had failed to beat in 17 previous attempts, Mark Hughes and Michael Green came close to making the margin even more emphatic for Eastleigh in the final moments.