First-half goals from Callum Lang and Watt gave Motherwell a cushion when Livingston played with the storm-force winds at their back, and the visitors survived a challenging second half without major scare.
Storm Aiden brought winds of about 50 miles per hour to West Lothian and Motherwell stood up to the aerial test after dominating the opening period.
It was Motherwell's fourth win in five Scottish Premiership matches and their first league win at the Tony Macaroni Arena since Livingston regained their top-flight status in 2018.
Motherwell kept the ball on the ground as much as possible with the wind behind them and they shaded the early proceedings, although Jay Emmanuel-Thomas was not far away with an ambitious effort for the hosts.
Well took the lead midway through the half when Devante Cole played the ball through the left channel for Lang, who turned inside his man and showed good composure to slot home.
At the other end, Jason Holt shot straight at Aaron Chapman, who threw the ball 15 yards inside the Livingston half rather than kick from his hand.
Motherwell continued to look more threatening. Lang sent Nathan McGinley away down the left and his cross was tipped over by Max Stryjek, who had kept his place after replacing on-loan Rangers goalkeeper Robby McCrorie for last weekend's 2-0 defeat at Ibrox.
Motherwell had a great chance on the break after clearing a Livingston free-kick. Liam Polworth sent Cole down the left channel but the forward curled well over with team-mates in support.
The visitors got their second just before switching ends. After Jon Guthrie had tripped Allan Campbell right on the edge of the box, Stryjek could not hold Mark O'Hara's free-kick and Watt was first to the rebound.
Motherwell boss Stephen Robinson brought on target man Jordan White at half-time in place of Cole, probably as much to help with defensive set-pieces as his side no doubt braced themselves for an aerial bombardment.
It mainly came in the form of long throws from Ciaron Brown but Motherwell, with Declan Gallagher and O'Hara impressive in central defence, were resolute. Livi did waste some chances to test Motherwell with crosses, with several going behind or into Chapman's hands.
Livi continued to get the ball in the box. Chapman tipped a long throw away under pressure, substitute Scott Tiffoney twice threatened with headers and curled a deflected effort wide.
But Well stood firm and Watt forced a diving save from Stryjek in a stoppage-time breakaway.