The Cobblers are in the third round of the FA Cup for the first time in six years after avoiding a potential embarrassment at the hands of the Dr Martens Premier side.
Two late goals from substitute Marc Richards, his first for the club, gave the scoreline an exaggerated look, as Frank Gregan's side put up a stirring display, despite playing the final 54 minutes with ten men.
The cup tie swung on the eagle-eyed ability of a linesman to spot Mark McKeever diving across goal and punching clear a ball from the goal-line, seconds after Stuart Jones had tipped a Josh Low drive onto the crossbar.
With McKeever sent packing by referee Mark Cooper and Martin Smith scoring his seventh goal of the season from the penalty spot, the minnows were on a slippery slope from which they never recovered.
Yet they had begun the match as though an upset was on the cards, Matthew Rose's third minute header needing urgent attention from Lee Harper in the home goal while former West Ham and Celtic veteran Stuart Slater curled a dangerous free-kick inches wide.
Northampton made heavy weather of the contest, but once Low had beaten Jones in the 64th minute with a 15-yard shot, the writing was on the wall.
Richards made it 3-0 13 minutes from time when he tucked the ball past Jones after running onto a pass from Des Lyttle only for Weston skipper Billy Clark to grab a deserved consolation with a looping header from a Slater free-kick three minutes later.
Richards had the last laugh in injury time with his second although the part-timers left the field with their heads held high.
Weston manager Gregan said: "The annoying thing is we don't know how well we would have done with 11 players for the whole of the game.
"We created some good openings early on but were left to rue the chances we missed." Northampton's director of football John Deehan, said: "We weren't happy that we didn't dominate the game more but there are no easy ties these days." He added: "You have to be realistic about what is possible for a club like ours in the FA Cup but it would be a fantastic achievement if we could get to the fifth round. Let's see what comes out of the hat."