Motherwell took the lead inside two minutes when Callum Slattery collected a loose ball, flicked it up and unleashed a 25-yard strike into the top corner.
County came back into it and Regan Charles-Cook scored a deserved equaliser six minutes before half-time when his dangerous cross ended up in the net.
Both sides hit the crossbar after the break and County had the bulk of the chances.
But Graham Alexander's team have developed a habit of grinding out results and they secured three points when Watt latched onto a ball over the top and despatched his fifth goal of the season.
Well had briefly threatened to build on their early goal. Kaiyne Woolery's dangerous cross was cleared and Watt fired over from 22 yards.
County soon began to get a foothold. Blair Spittal saw three shots saved and hit one over the bar with the openings getting better and better.
Alexander was booked after a couple of referee Craig Napier's decisions riled him and County continued to dominate possession, although Motherwell defender Juhani Ojala had a header saved.
The equaliser came when Charles-Cook curled a cross in between the Motherwell back four and Liam Kelly. Jordan White and Ross Callachan could not quite get a touch on it but they did not need to as the ball nestled in the far corner.
Watt sent a header against the bar early in the second half after Stephen O'Donnell put the ball back into the box following Ashley Maynard-Brewer's stop from Woolery.
The visitors were soon back on the front foot. Joseph Hungbo had replaced Charles-Cook at half-time and the winger hit the bar after a well-worked move.
The on-loan Watford player looked poised to head home moments later until Nathan McGinley threw himself in the way. The grounded left-back also managed to prevent Callachan scoring before Napier ended the goalmouth scramble by awarding a free-kick to Motherwell.
The home team were struggling to work the ball forward without resorting to hitting it long and County were looking the more likely to score. Jack Burroughs and Hungbo threatened from long range and White headed over.
McGinley engineered a promising break with some aggressive defending and a strong run but he opted to shoot instead of setting up one of his forwards and the shot was blocked.
Spittal had a free-kick saved before Motherwell again turned defence into attack thanks to a brilliant diagonal ball from Ricki Lamie, who had returned to the team in place of the absent Bevis Mugabi.
The pass put Watt in the clear down the right channel and he fired the ball underneath Maynard-Brewer.
Alexander brought on defender Jake Carroll for striker Kevin Van Veen and Motherwell saw out the win without incident.