Shane Tudor and Gabriel Zakuani scored the goals for Leyton Orient as they kept the home side at bay.
Bury started the brighter of the two sides and came close to taking the lead in the 15th minute.
Stalwart midfielder David Flitcroft saw a shot charged down and then a minute later Dwayne Mattis, last year's player-of-the-season, shot agonisingly wide of the left-hand post.
The home side continued to pressure, but were hit by a sucker-punch when Lee Steele saw a shot parried by goalkeeper Neil Edwards. Tudor was on hand to calmly slot the ball home for an early advantage.
Defender Paul Scott did well to clear off the line in the 35th minute after some more trickery from Steele, before Edwards saved a long-range effort from Donny Barnard.
The resumption saw former Stockport County full-back John Hardiker have a shot saved by opposing goalkeeper Glyn Garner, making his first return to Gigg Lane since last year's departure.
But Zakuani doubled the visitors advantage in the 72nd minute, jumping well to head home a corner from Matthew Lockwood which Edwards failed to clear with his punch.
Smart hit back for Barrow's men with 13 minutes remaining, but Darlington clung on for the points, although Garner saved well from a Brian Barry-Murphy free-kick.