The midfielder slotted home the only goal of the game in the 27th minute to secure three massive points in Town's battle against relegation from Sky Bet League One.
All of the Cobblers' rivals at the bottom of the table were beaten, and the win means Town move up to 19th in the table, three points above the drop zone.
Boss Jon Brady made one change to the team that lost at AFC Wimbledon last Saturday, with Ryan Edmondson returning in attack in place of Mark Marshall who dropped to the bench.
The match was preceded by an emotional minute's silence for former Cobbler and Shrews defender Lee Collins, who died on Wednesday aged just 32.
Both sets of players also wore black arms bands as a mark of respect.
After a cagey opening where neither side create any clear-cut openings, it was the Cobblers who almost opened the scoring on 25 minutes.
Watson broke down the left and his inswinging cross was flicked on to the far post where Hoskins got on the end of it.
The Town man was on the stretch and it was a tight angle, but Hoskins got a good connection, only for Matija Sarkic in the Shrews goal to save low to his right.
The Cobblers suddenly had a spring in their step, and they were ahead a couple of minutes later through top-scorer Watson.
Peter Kioso delivered a low cross from the right, and Watson met it just inside the area and expertly side-footed the ball first time into the top left-hand corner with his left foot. It was a quality finish.
Shrewsbury, a disallowed effort for a tight offside apart, had offered very little in attack, but they passed up an excellent chance of an equaliser on 39 minutes.
The livewire Harry Chapman should have done better, but he headed over Nat Ogbeta's teasing left-wing cross from just eight yards out.
The Cobblers then had a chance to double their lead on the stroke of half-time.
Hoskins did well to latch on to Edmondson's looped pass over the top and he got his shot off under pressure, but he fired the ball straight at Sarkic, who saved.
The second half saw Shrewsbury up their intensity, but they struggled to create any chances of note.
It did mean the Cobblers struggled to get any sort of attacking momentum though as the game turned very scrappy.
It was the 75th minute before either keeper was called into serious action, and then it was easay for Jonathan Mitchell as he plucked Chapman's effort from 25 yards out of the air.
At the other end Kioso tried his luck from the right corner of the area, and although the shot had a sting, it was straight at Sarkic.
The game got more and more scrappy as the Cobblers defended more and more desperately, but they did their job as Shrews were unable to create any sort of opening and Town's players celebrated what could be a huge win.
Cobblers: Mitchell, Kioso, Horsfall, L Jones, Mills, Morris, McWilliams, Watson, Hoskins, Edmondson (79m, Sheehan), Rose. Subs not used: Harriman, Bolger, A Jones, Chukwuemeka, Berry,
Shrewsbury: Sarkic, Pierre, Shaun-Williams (89m, Goss), Norburn (89m, Whalley), Vela, Ogbeta, Love, Main (79m, Pyke), Ebanks-Landell, Davis, Chapman (89m, Edwards). Subs not use: Burgoyne, Sears, Pennington
Referee: Ross Joyce