Brown's long diagonal ball set up Odsonne Edouard to double Celtic's lead in the 24th minute.
David Turnbull had netted the opener 60 seconds earlier after loan goalkeeper Zdenek Zlamal had been controversially penalised for picking up a passback.
Zlamal had been pitched in for an unexpected Saints debut with the visitors without more players amid what manager Callum Davidson termed an ongoing issue with Covid.
Aberdeen-bound Brown arrived to a welcoming party of fans outside the main entrance and there were several banners in his honour in the stands as he led the team out for the final time at Celtic Park.
The 35-year-old was presented with a silver salver by outgoing chief executive Peter Lawwell before the game to mark his 14 years of service, and got a standing ovation from the directors box and substitutes after being replaced by Ismaila Soro.
There was a debut as well as a swansong as Adam Montgomery lined up at left-back for Celtic. The 18-year-old delivered a composed and confident display in possession as well as a goal-saving intervention.
Saints had not trained since their Scottish Cup semi-final triumph after losing four players to self-isolation for Sunday's win over St Mirren. They had further absentees at Celtic Park with Zander Clark, Liam Gordon, Ali McCann and Michael O'Halloran missing from their 16-man squad.
With Charlie Gilmour making his first start, the visitors started well. The stretching Chris Kane could not steer the ball on target after James Brown headed across the face of goal.
The game turned on a moment of controversy when Zlamal picked up after wing-back Brown knocked the ball backwards. Davidson could not believe referee Bobby Madden had deemed the action deliberate and his mood worsened when Turnbull drove inside the near post from 16 yards following the indirect free-kick.
Scott Brown soon had his moment as Edouard finished in style from his long pass, the French striker producing a brilliant take, cutting inside Shaun Rooney and dinking over Zlamal.
Vasilis Barkas was making his first Celtic appearance since January after Scott Bain injured a finger and the Greek goalkeeper produced a decent save at full stretch to deny Kane.
Celtic set about adding to their lead. Turnbull curled wide, Rooney cleared off the line from Ryan Christie and Zlamal made saves from Edouard and Christie.
Montgomery cleared off the line from Craig Conway before making way for Greg Taylor when Brown was taken off in the 63rd minute.
The remainder of the chances came at the other end. Zlamal denied Edouard at close range before the Frenchman inadvertently blocked Mohamed Elyounoussi's follow-up.
The loan keeper produced good saves from substitutes Taylor and Leigh Griffiths but was beaten by Ajer's heavily-deflected effort from inside the box in the 79th minute.
Dembele raced on to Turnbull's through ball and found the net from a tight angle six minutes later.
With Livingston losing 3-0 to Rangers, the scoreline was enough to keep Saints above the West Lothian side on goal difference ahead of their shoot-out for fifth place on Saturday.