The game was drifting when the 18-year-old striker replaced Pawel Brozek in the 59th minute but it came to life when he volleyed in a Victor Wanyama cross three minutes later from a tight angle.
With the travelling support still celebrating, he drove low past Well goalkeeper Darren Randolph from the edge of the box.
Hoops wing-back Cha Du-ri headed in a third to deal a blow to the home side's Champions League aspirations, the Steelmen now only three points ahead of fourth-placed Dundee United having played the same amount of games.
But this game was all about the former Airdrie United player who would have given manager Neil Lennon immense satisfaction as he sat in the stand serving the first of his two-match touchline ban for criticising referee Willie Collum after last month's Scottish Communities League Cup final defeat by Kilmarnock at Hampden.
Lennon had handed Watt a place on the bench in a much-changed line-up.
Polish striker Brozek, on loan from Trabzonspor until the end of the season, made his first start while number two keeper and fellow Pole Lukasz Zaluska made a rare appearance in goals.
Full-backs Cha and Emilio Izaguirre, stopper Thomas Rogne and midfielder Wanyama also came in with Kelvin Wilson, Scott Brown, Georgios Samaras, Glenn Loovens, Fraser Forster and Mikael Lustig dropped out, with only the latter two starting on the bench.
Motherwell defender Stevie Hammell and midfielder Steve Jennings both returned from a ban while Jonathan Page and Keith Lasley were missing through suspension, as Omar Daley came in for Henrik Ojamaa, who started on the bench.
The visitors, captained for the day by striker Gary Hooper, started impressively.
Midfielder Kris Commons flashed a long-distance drive wide of the target before Brozek headed a Cha cross over the bar.
The Steelmen were pinned back by the width Izaguirre and Cha were giving Celtic down the flanks.
Motherwell's first real attack came midway through the first half when a flick from burly striker Michael Higdon set winger Chris Humphrey free but his drive drew a fine save from Zaluska.
The Parkhead side still controlled possession but in a swift Fir Park counter, the Celtic keeper had to make another save from Humphrey.
However, in the 34th minute Brozek passed up on a great chance to open the scoring after Wanyama had picked him out but the Hoops striker stabbed awkwardly at the ball which trundled past Randolph's right-hand post.
Five minutes from the interval Celtic midfielder Ki Sung-yueng limped off to be replaced by Paddy McCourt, who moments later lost possession giving Daley the chance to test Zaluska with an angled drive.
The first half ended with Randolph busy, the keeper making a save from a well-timed volley from Commons and a more fortuitous block from Brozek after the Poland international had made space for himself in the Fir Park box.
Lennon sat in the directors' box during the interval, unable to go down to the dressing room, and he watched Motherwell start the second half with some purpose.
Four minutes after the restart, Daley set up Nicky Law whose powerful drive from the edge of the box spun wide, the midfielder's second effort moments later guided better but easily saved by Zaluska.
But Celtic soon reasserted themselves and in the 55th minute McCourt split open the Fir Park defence with a great pass to find Cha racing into the Motherwell box but the South Korean's weak cut-back saw the ball was easily scooped up Randolph.
The disappointing Brozek was replaced by Watt, before Daley was hooked for Ojamaa.
Watt then made himself an almost instant hero with the Hoops supporters when he fired Celtic ahead by returning a deep Wanyama cross back towards the goal with Randolph unable to keep it out.
Three minutes later he was set up by Joe Ledley before drilling the pass low past Randolph from the edge of the box to complete a dream debut.
Watt's timely intervention had brought the Hoops fans to life and the obvious potential for a hat-trick was in the minds of everyone in the stadium.
However, in the 82nd minute it was Cha who got Celtic's third goal, heading in a deep McCourt cross from close range.
The travelling fans were now in full voice as they chanted for Lennon, who must have watched with some satisfaction from his seat in the stand.
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