Watson struck either side of half-time - Town's first home goals of 2021 - to clinch three sorely-needed points and take Cobblers away from the bottom of the table and up to 22nd place, giving them fresh hope of avoiding relegation.
Watson's first goal was a fabulous strike from outside the box shortly before the break and the in-form midfielder then had his fourth in four games with a close-range finish midway through the second-half, securing Jon Brady's first win in caretaker charge.
New signing Alex Jones was on the bench as Brady made three changes with Fraser Horsfall, Jack Sowerby and Shaun McWilliams, who made his 100th Cobblers appearance, all back in the starting line-up.
The game started sluggishly but spluttered into life with a flurry of end-to-end action 12 minutes in when Mickel Miller was unable to pick out a team-mate from a dangerous crossing position before Plymouth broke at pace and Jonathan Mitchell reacted smartly to deny Luke Jephcott.
Miller was a bright spark for the Cobblers and his cross-shot had to be beaten away by Michael Cooper, but the first-half, whilst closely-contested, was short on goalmouth action.
Plymouth played some neat football without creating too many goalscoring opportunities and Town also had their moments, most notably after 36 minutes when Miller's sweet first-time volley was turned behind by Cooper.
But the home side's best moment was saved until five minutes before the break when Ryan Edmondson teed up Watson and he did the rest, drilling a fabulous effort in off the post from 20 yards.
That gave Town a rare and precious half-time lead but Plymouth seized almost total control of possession in the second-half as they patiently tried to work an opening.
Horsfall and Joseph Mills got vital touches on dangerous crosses but Town went up the other end midway through the half and crucially gave themselves breathing space.
It was a scruffier goal this time as the ball bounced around in Plymouth's box at a corner but Cobblers - nor Watson - cared as he turned into the net, doubling Town's lead.
Plymouth continued to take their time in trying to break down Northampton but the hosts worked extremely hard to keep them at bay, with Horsfall particularly impressive in defence.
Argyle's night worsened in stoppage-time when Sam Woods was shown a straight red card for an ugly looking challenge on Ricky Korboa, and then Mitchell preserved his clean sheet with a superb stop from Jepchott with virtually the last act of the game.
Cobblers: Mitchell, Kioso, Jones, Horsfall, Mills (c), McWilliams, Sowerby, Watson, Hoskins, Miller (Rose 72), Edmondson (Korboa 86)
Subs not used: Arnold, Harriman, Morris, Marshall, Jones
Plymouth: Cooper, Watts, Aimson, Edwards (c) (Moore 67), Fornah (Lolos 82), Lewis (Grant 67), Woods, Camara (Reeves 67), Hardie (Ennis 67), Mayor, Jephcott
Subs not used: McCormick, MacLeod
Referee: Graham Salisbury