The first chance of the game came to the home side on three minutes, as Will Dennis was called into action with the Rebels keeper making himself big to deny Leone Gravata the opening goal.
Eastbourne hit the bar on six minutes after a corner was put into the box with Bradley Barry rising above everyone and heading the ball onto the top of the crossbar.
Eastbourne continued their early pressure on 15 minutes as Luke Pearce’s shot from close range was superbly cut out by Lee Togwell.
On 22, The Rebels gave the ball away cheaply in midfield as Gravata ran at the Slough defence with the shot straight down Will Dennis’s throat.
Shiloh Remy almost opened the scoring for the hosts on 33 minutes after the ball was played into the Eastbourne midfielder, who shot first time on the half-volley with Dennis equal to the shot.
Slough started the second half stronger as on 51, Ogo Obi did brilliantly to drive into the box; he turned and put a cross into the six yard area with Elliot Benyon getting onto the end of it but the header was straight at Lee Worgan.
Eastbourne then took the lead four minutes later after Remy crossed the ball towards the far post, with the ball finding Charlie Walker who, having moments earlier hit the post with a header, made no mistake this time as he guided the ball past Dennis.
Walker nearly doubled Eastbourne’s lead on 67 after the captain found himself one-on-one with Dennis, but the Rebels keeper pulled off a big save at full stretch to keep Slough in the match.
Eastbourne struck the woodwork for the third time on 75 minutes after James Hammond attempted to score with a free kick from distance. The effort was dipping, but Dennis did brilliantly to tip the ball onto the crossbar.
Miquel Scarlett was the last player to strike the post two minutes from time as his effort from the edge of the area struck the outside of the post and thankfully stayed away from the Slough net.
But Norman Wabo wrapped up the three points for Eastbourne in stoppage time. Morgan Williams did some good work down the right hand side and drilled the ball towards Wabo, who made no mistake from close range.
Eastbourne Borough Lineup
Lee Worgan, Jack Burchell, Bradley Barry, Ryan Bartley, Jayden Perez, James Hammond, Charlie Walker, Shiloh Remy, Luke Pearce, Leone Gravata, Sam Oguntayo. Substitutes: Greg Luer, Morgan Williams, Norman Wabo, Miquel Scarlett, Jay Beckford.