Darlington mastered difficult conditions to record a deserved win.
Guylain Ndumbu-Nsungu put the Quakers ahead after 44 minutes after Shelton Martis' free-kick hit a post and the striker was on hand to score.
Darlington went two up after 53 minutes when Neil Wainwright crossed for Simon Johnson to score at the far post.
Barnet briefly threatened at the end when Ben Strevens fired a consolation seconds before the final whistle, but Darlington merited the victory.
Darlington: Bossu, Close, Martis, Clarke, Kendrick, Wainwright, Keltie, Dickman, Johnson (Logan 63), Ndumbu-Nsungu (Peacock 83), Kandol.
Subs Not Used: Appleby, Sodje, Webster.
Booked: Ndumbu-Nsungu, Logan.
Goals: Ndumbu-Nsungu 44, Johnson 53.
Barnet: Flitney, Hendon (Batt 59), Yakubu, King, Clist, Strevens, Lee, Vernazza, Bailey, Norville (Sinclair 69), Hatch.
Subs Not Used: Tynan, Bowditch, Gross.
Booked: Bailey.
Goals: Strevens 90.
Att: 2,905
Ref: R Booth (Nottinghamshire).