A first half strike from Stuart Beavon the difference as good pressure towards the end was not enough.
Wrexham started brightly and Beavon played Holroyd seven minutes in, but the forward was flagged offside.
Jordan Maguire-Drew had the first clear cut opportunity of the game when he was played through, he his effort was off target.
Akil Wright was then almost in on 25 minutes, when Mark Cousins fumbled a high ball under pressure from Holroyd, but the ball was turned behind.
Shortly before the break, Wrexham took the lead through Beavon when he slotted home into the right corner.
Barnet's first shot on target came in the second half when Wes Fonguck latched on to a mistake by the Wrexham defence but Lainton saved well.
Wrexham looked to push for a second, however, and Maguire-Drew’s cross found Beavon on 54 minutes but his effort was off target.
On the hour mark, Barnet began to push and stretch the Wrexham defence further back.
The Bees continued to deal with everything thrown their way and with only a one goal defecit, they were very much still in the game.
Substitute Nana Kyei’s low shot was saved by Lainton on 80 minutes.
A foul on the edge of the area gave Dan Sparkes a chance at a goal but his effort and follow up shot were both blocked..
Robson’s header at the end of regulation time from Kyei's cross looped agonisingly over the bar.
Barnet: Cousins, Alexander, Robson, Jules, Tutonda, Adams, Mason-Clark (Kyei 77), Taylor, Fonguck, Sparkes, Barham.
Subs not used: Matrevics, Elito, Payne