Beckford had remarkably gone seven games without a goal prior to Monday's first-round replay, but he ended that spell with a blistering hat-trick at Sixfields.
The Whites now tackle Blue Square Premier outfit Histon in the second round of the competition following their surprise 1-0 success over Swindon.
Fleet-footed striker Beckford opened his account on 13 minutes after volleying home superbly when David Prutton flicked on Lubomir Michalik's long-ball.
Leeds doubled their advantage when Jonathan Howson's drive was deflected past Northampton keeper Frank Fielding by his own defender Mark Hughes.
Fielding's mistake from Ben Parker's long-range drive put Leeds 3-0 up, but the home side responded with Jason Crowe pulling one back to give the home fans hope.
However, the visitors restored their three-goal cushion before the break when Beckford got the better of Hughes before sending a delightful right-footed curler past Fielding and in off the far post.
The former Wealdstone United striker secured his hat-trick with 55 minutes played when he latched onto Neil Kilkenny's through-ball to slide the ball past Fielding.
Northampton ended the match on a positive note when Crowe was perfectly placed to nod home Liam Dolman's flick-on, but it was nothing more than a consolation.