It was a Group G clash that the Hoops were not supposed to win but the east end of Glasgow exploded in the 21st minute when midfielder Wanyama headed in from a Charlie Mulgrew corner.
It was backs-to-the-wall stuff for most of the game for the home side as the star-studded visitors hit the woodwork twice and goalkeeper Fraser Forster pulled off a number of vital saves.
But in the 83rd minute 18-year-old Watt, on for Mikael Lustig to make his debut in the competition, raced away to score the second, leaving Lionel Messi's late strike a consolation.
Celtic, with seven points from four games, two less than leaders Barca, travel to Benfica before hosting Spartak Moscow in their final group fixture and will surely fancy their chances of reaching the knockout stages of the competition.
Regardless of the Scottish champions' progress in Europe, the night will live long in the memory of the 60,000 who made their way to Celtic Park on a sodden night.
There had been some frenzied speculation as to who manager Neil Lennon would have available.
As it transpired, the former Celtic skipper suffered an unforeseen blow when skipper Scott Brown dropped out through illness.
Georgios Samaras, however, recovered from an ankle injury sustained in the Nou Camp last month but as expected Gary Hooper, Emilio Izaguirre, Thomas Rogne, James Forrest and Lassad were all missing through various ailments, with Lustig, Efe Ambrose and Miku also coming into the side which drew 2-2 with Dundee United at Tannadice on Sunday.
Gerard Pique was added to Barcelona's squad after returning from a foot injury but he started on the bench, although the Catalan giants paraded their big guns in Messi, Andres Iniesta and Xavi.
There was a card display before the kick-off to celebrate Celtic's 125th anniversary which added to the terrific atmosphere at kick-off time.
Stand-in captain Samaras led the line for the home side supported by Miku, but like the rest of the Celtic side they were soon chasing Barcelona around looking to gain possession.
Alexis Sanchez and Messi had early efforts for the visitors who soon fell into their familiar, dominating pattern.
Barca were perhaps fortunate in the 18th minute when midfielder Alex Song, earlier booked for a challenge on Miku, fouled the same player and received no further punishment from referee Bjorn Kuipers.
However, the home fans were unconcerned about that when Wanyama gave them a sensational lead.
The Catalans had conceded from a set-piece in the Nou Camp and looked equally ropey when the Kenyan rose above rookie defender Marc Bartra at the back post to bullet in Mulgrew's corner from the right.
Tito Vilanova's side went up a couple of gears and hemmed Celtic in at the edge of their own penalty area and in the 29th minute Iniesta set up Messi and his first-time strike clipped the bar on the way over.
Celtic were under relentless pressure and in the 33rd minute Jordi Alba fired the ball across the six-yard box after being played in by Alexis Sanchez but found no takers.
Moments later, the Chilean forward glanced a header from a Dani Alves cross off the post and Celtic Park breathed again.
Five minutes from the break, with the home side defending with equal measures of diligence and desperation, Forster thwarted Messi as he attempted to get on the end of another Jordi Alba cross.
The second half began in similar style to the first with Barca taking control, switching from left to right and back again.
Alves flashed the ball across Forster's six-yard box but again there was no one in blue and red to take advantage.
In the 56th minute Messi tried to beat Forster with a left-footed curler from 20 yards but the England squad member was equal to the task and soon made a good save from Alexis Sanchez's angled drive after Messi had split open the defence.
Despite all their possession, Barcelona looked vulnerable at the back and Mulgrew might have punished them in the 63rd minute when he had a sight of goal but Bartra managed to block. Song was allowed another foul on Miku to the anger of the Celtic fans, before he was replaced by Cesc Fabregas, with Pique on for Bartra and Watt on for Lustig.
Before the substitutions, Forster flew through the air to make a great save from Messi.
Beram Kayal replaced Samaras with 10 minutes remaining before Forster made yet another save from Messi, this time from a 25-yard free-kick.
But there was much more drama when Watt escaped the Barca defence and Xavi from Forster's long ball before driving a right-footed shot past Victor Valdes to double Celtic's lead.
There was bedlam inside Parkhead which gave way to some nerves when Messi scored from close range in the 90th minute after Forster had parried a strike from Pedro, but it was to be the home side's night, and a famous night at that.
