Slough started quickly, with Barney breaking down the left and trying to cut the ball back for Warren Harris, but the ball was cleared.
Just two minutes later, Dan Hicks clipped a pass in for Lewis Putman, but his volley hit the side netting and the winger was flagged for offside.
Linnets striker John Sands twice went close, first having a header well tipped away by Mark Scott, before hitting the post with a good effort from outside the box.
Play settled down but Sands threatened again as he flicked an audacious effort just over and Liam Hurst went closer still, seeing an attempted lob from the edge of the box cleared off the line by Harris after a corner was cleared.
Somewhat against the run of play, Slough took a 31st minute lead through Barney as he coolly slipped the ball through the legs of Alex Street from Gurkan Gokmen's pass.
Neat link-up play between Harris and Putman almost resulted in a second goal as the latter played a low ball across goal, but Barney put it behind for a goal kick from six yards under pressure.
The Rebels took their lead into half time with the visitors firing off target three times from outside the box, and in the second half Lynn enjoyed much of the possession in the without troubling Scott unduly.
Two minutes in, Nathan Webb nodded the ball through for Barney, who scuffed his shot but still forced Street to dive on the ball with Harris lurking at the back post.
Webb later had an effort ruled out from offside, whilst for the Linnets Liam Hurst's screamer from distance flew just wide of the post. Lewis Putman also saw a low shot held by Street.
Lynn cranked up the pressure in the closing stages but were repelled by the Rebels. Scott blocked a Sands strike with his legs and comfortably saved from Lee Stevenson, who also blasted a free kick wastefully over the bar from a dangerous position.
Three minutes remained when the visitors put a free header over the bar following a corner, and Lynn were made to pay for their missed chances as Barney confirmed the win two minutes into added time, finishing from close range after Putman beat his man and pulled the ball across goal.
King's Lynn Town Line up
Alex Street, Ryan Fryatt (c), Jordan Yong, Matthew Castellan, Kern Miller, Lee Smith (Liam Fryatt 65), Michael Clunan (Dylan Edge 76), Liam Hurst (Toby Hilliard 71), Sam Mulready, John Sands, Lee Stevenson. Unused subs: Jacek Zielonka, Jack Friend.