Paddy McCourt's first-half strike separated the sides but the Hoops should have had at least another three in a one-sided first half.
The home side looked less impressive after the break but while they never looked like losing they received a late blow in the 89th minute when skipper Brown, just back in the team after a lengthy injury and booked earlier for a foul on Chris Humphrey, received a second yellow for a foul on Stevie Hammell.
Celtic, who also saw substitute Daryl Murphy limp off injured, are now one point clear of Rangers, albeit having played two games more than the champions, but manager Neil Lennon will have to look for a replacement for the hot-headed Brown.
For their part, Motherwell, expected to confirm Stuart McCall as their new manager in the morning, were poor again and will need some reorganising by the former Scotland international.
Lennon rang the changes for the visit of the Fir Park men.
With Ki Sung-yueng and Cha Du-Ri away at the Asian Cup with South Korea, Olivier Kapo made his first start for Celtic with James Forrest, Joe Ledley, Mark Wilson and Anthony Stokes also coming in to the starting line-up.
The visitors showed one change to the side which lost 4-1 at home to Rangers at the weekend with caretaker boss Gordon Young bringing in Marc Fitzpatrick for Nick Blackman, who dropped to the bench.
Lennon had Kapo in attack alongside Stokes, with what was yet another strike-pairing, but it was defender Emilio Izaguirre who had a great chance in the second minute when Well keeper Darren Randolph spilled a low Ledley drive.
However, despite facing an open goal less than six yards out, the Honduran left-back somehow blasted over the bar.
The Lanarkshire men were simply looking to regroup after their painful home defeat by Rangers but they were under siege as Celtic swarmed around their box.
Kapo had the ball in the net in the 12th minute only to be ruled offside, seconds before McCourt headed a Forrest cross wide of the target.
Motherwell looked bereft of hope, belief and confidence and it was not surprise when, in the 27th minute, Celtic took the lead with the inspirational McCourt breaking the deadlock.
After yet another Hoops corner had the Fir Park men pinned back in to their own penalty area, the Northern Ireland international took a pass from Wilson, drifted in to the box and almost casually curled a right-footed shot from 16 yards in to the far corner past Randolph.
Six minutes later, with the Well defence all over the place, Ledley was sent clear by Stokes and his low left-footed shot from 16 yards sped past Randolph but also the far post.
Kapo was replaced by Daryl Murphy in the 38th minute but Celtic cruised to the half-time whistle in complete control.
There was blow for the Hoops in the 51st minute when substitute Murphy limped off injured to be replaced by midfielder Beram Kayal.
The change seemed to disrupt Lennon's side and for the first time in the game Motherwell began to string a few passes together.
In the 58th minute, as the home defence came under some pressure, Fitzpatrick got on the end of a Chris Humphrey cross but failed to connect properly from six yards and the ball bounced in to the arms of Celtic keeper Fraser Forster.
The game turned rather nasty as the tackles began to fly in, the home fans incensed by those which were perpetrated on their players.
Blackman replaced Jamie Murphy in the 76th minute as the visitors sensed there might be a point there for the taking.
There were some nerves in the final stages and in the 89th minute Brown, booked earlier, was sent packing by referee Stephen Finnie after a challenge on Hammell to take the shine of Celtic going top of the table.