A statue in honour of former captain and manager Billy McNeill was unveiled outside Parkhead before the game and the impressive sculpture of a Hoops giant holding aloft the European Cup won in 1967 seemed to cast a shadow over Ronnie Deila's side as they fell to Mark McGhee's side.
Celtic were well off the pace despite taking the lead in the 49th minute with a deflected Nir Bitton shot, who had been set up by Rogic.
Well striker Louis Moult levelled four minutes later before scoring from the spot in the 59th minute after Celtic stopper Jozo Simunovic tripped former Celt McDonald.
The win was Well's first away victory over Celtic since 2008.
It was a defeat which allowed Aberdeen to move within a point of the champions, albeit having played a game more, after they thumped Kilmarnock 4-0 at Rugby Park.
Aberdeen's fourth straight victory was the ideal way to celebrate Derek McInnes's 100th league game in charge of the Pittodrie men.
First-half headers from Niall McGinn and Adam Rooney put Aberdeen completely in command and Jonny Hayes made it three just 16 seconds into the second half. Shay Logan then wrapped up the win when he stroked home with 10 minutes left.
Second-bottom Killie's troubles had offered Dundee United hope as they look to claw their way back up from the foot of the table.
But they were dashed when Lewis Horner bundled home an injury-time equaliser for Inverness as Caley Thistle claimed a point with a 2-2 Highlands draw - leaving the Taysiders seven points adrift.
Horner turned in Carl Tremarco's cross in the second minute of added-on time to cancel out Billy McKay's late strike as the Northern Ireland international made his return to the Caledonian Stadium a year after quitting for Wigan Athletic. John Rankin had given Mixu Paatelainen's side the lead 26 minutes in but a smart finish from Liam Polworth straight after half-time restored parity.
In Perth, Steven MacLean missed a late penalty and a chance to ease St Johnstone past 10-man Heart of Midlothian into third place.
Hearts went a man down after 27 minutes when Juanma Delgado was shown a straight red card for an attempted head-butt on David Wotherspoon but St Johnstone could not take advantage as the match finished goalless.
It could have been a different story for the hosts, though, had they made the most of the opportunity they were given 12 minutes from time when Hearts substitute Miguel Pallardo shoved Perth replacement Chris Kane in the box. MacLean, though, lashed his spot-kick attempt high over the bar as the Jambos held on for a point.
Kane Hemmings's first-half hat-trick, meanwhile, helped Dundee celebrate their first top-flight win since early October.
It took Paul Hartley's side just 25 minutes to sweep Hamilton Academical aside at Dens Park as Hemmings added his treble to Greg Stewart's 19th-minute strike.
It was game over by then as the Dark Blues claimed a win which moves them into the top six.
Partick Thistle notched up a fourth consecutive top-flight win for the first time in almost 40 years after securing a narrow 1-0 victory over Ross County.
The last time the Glasgow outfit achieved the feat in Scotland's top division was 1977 - but they matched that record thanks to an own goal by Ross County skipper Andrew Davies.