Barnet booked their place in the second round of the FA Cup despite seeing Connor Stevens sent off on his debut.
Idris Kanu curled an early effort just wide, then picked out Sam Beard at the far post to head the ball back across goal, but Nicke Kabamba was beaten to it by the home keeper.
The visitors opened the scoring on 14 minutes, though, as Finley Potter produced a great cross to find Kabamba to head home from close range.
And a great block by Stevens, who had been shown an early yellow card, denied Curzon an equaliser moments later.
Zak Brunt's fine pass found Kanu for a low drive from the edge of the box which was well saved on 21 minutes, before good work from Callum Stead found Kanu, but he was unable to get a shot on target.
But the hosts then got a brief sight of goal but sent an attempt from the edge of the box over the target on 39 minutes.
Bees were reduced to 10 men soon after when Stevens was dismissed for a second bookable offence and they had a let-off during stoppage time as Potter cleared off the line to deny the hosts an equaliser.
Potter and Beard made way for Danny Collinge and Jordan Cropper during the break, as Dean Brennan rung the changes.
But the visitors needed Laurie Walker to make a superb one-handed save to keep out a long-distance effort on 54 minutes to preserve his side's lead.
Kanu was booked just before the hour mark, with Curzon seeing an effort ruled out by a linesman's flag seconds later.
And Walker had to make another vital save, clawing the ball away from danger when Curzon played it across the face of goal.
The hosts fired another effort from distance over the crossbar on 71 minutes, but Stead then rounded the Curzon keeper only to hit the post.
And one goal proved enough for Brennan's men to ensure their name would be in the hat for the draw on Monday.
Barnet: Walker, Okimo, Kabamba (Hooper 75), Stead (Armstrong 88), Kanu, Pritchard, Hartigan, Stevens, Beard (Collinge 46), Potter (Cropper 46), Brunt (Gorman 75). Unused subs: Gillmore, Senior.
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