The game started with a bang with Celtic's Mo Camara forced to clear a Richie Foran header off the line from a Steven Hammell corner and Well keeper Graeme Smith tipping a Maciej Zurawski header over.
Celtic eventually took the lead on 17 minutes when Zurawski pulled off the shoulder of Willie Kinniburgh to coolly slide the ball past the advancing Smith.
Well were never out of the game for the remainder of the half and got a deserved equaliser with four minutes to go to the break.
A Brian Kerr cross was met on the full volley by Jim Hamilton and his sweetly struck shot left Artur Boruc grasping at fresh air for one of the finest goal many will have seen.
At the start of the second half the third goal was always going to be crucial and it was the visitors scored it on 71 minutes.
Aiden McGeady, who had replaced the feckless Shunsuke Nakamura ten minutes earlier, hit a speculative shot from 20 yards and it hit Kevin McBride and spun away from the stranded Smith.
Two minutes later the home side carved out a superb chance but Scott McDonald could only head narrowly wide after combining with Foran to create the opportunity.
As the home side pushed forward for an equaliser the were caught on the break when McGeady skipped past McBride and Martyn Corrigan before sending an inch-perfect pass to Hartson to sidefoot past Smith from 15 yards out.
Despite a late flurry, Well were unable to find another goal that would have put pressure on the visitors in the closing minutes.