The 18-year-old, signed from Airdrie last year, replaced Pawel Brozek in the 59th minute and scored his first just over three minutes later when he volleyed in Victor Wanyama's cross.
Watt grabbed his second in the 66th minute, sending a low, angled drive into the bottom-left corner to mark a memorable start to his Celtic career.
South Korean international Cha Du-Ri rounded off the win when he headed in substitute Paddy McCourt's cross in the 83rd minute.
The win keeps the Hoops 18 points clear of second-placed Rangers ahead of next Sunday's Old Firm derby at Celtic Park.
At Fir Park, Celtic made six changes to the side that lost to Hearts last Sunday as manager Neil Lennon, who watched the match from the stands as he started a two-match touchline ban, rested several key players ahead of next week's match with fierce rivals Rangers.
Kris Commons was the first to try his luck but his long-range shot flew past the post before Wanyama watched his effort from 25 yards go the same way.
Lukasz Zaluska was making a rare start in goal for the Hoops and he had to be alert to stop Chris Humphrey's shot on the half-volley after the forward latched on to Michael Higdon's headed flick-on.
Humphrey had a go again moments later when he picked up a loose ball on the edge of the box and fired in a fierce strike that Zaluska got down smartly to stop.
On-loan Polish striker Brozek started his first game for Celtic and missed a golden chance to put the Hoops ahead in the 35th minute when Wanyama picked him out with a pass but he poked the ball well wide with only the goalkeeper to beat.
Darren Randolph came to the rescue for Motherwell on the stroke of half-time when he stopped Brozek's low drive with his feet after the striker found some space in the box.
Omar Daley created a chance for Nicky Law at the start of the second half but the midfielder sliced his shot wide.
Watt replaced the ineffectual Brozek in the 59th minute and he scored with almost his first touch in the 63rd minute.
Wanyama sent in a measured cross from the right and the teenager met it at the far post to send an angled volley under the legs of Randolph.
Watt got his second after Joe Ledley picked him out in the box with a pass and he took a touch before drilling an angled shot through the legs of Shaun Hutchinson and past Randolph.
Cha made sure of the three points when McCourt swung the ball into the box and the South Korean bulleted a header past the keeper from close range.
Source: AFP
Source: AFP