Both sides created chances in the first half but Giles Coke was shown a straight red card in stoppage time for a bad challenge on Scott Arfield.
Falkirk failed to take advantage of their numerical superiority, however, and remain bottom of the SPL, although they did force Motherwell goalkeeper John Ruddy into two impressive second-half saves.
Brown's appointment until at least the end of the season came less than an hour before kick-off and followed the earlier good news that Ruddy and Everton team-mate Lukas Jutkiewicz had extended their loan deals until the summer, although the striker had to settle for a place on the bench after having 12 stitches inserted in a thigh wound during Saturday's 2-0 win over St Mirren.
Motherwell controlled the early stages and Falkirk manager Eddie May, already missing Brian McLean through injury, was forced into a reshuffle of his defence when Pedro Pele was stretchered off in the 13th minute. Stewart Murdoch came on at right-back, with Jackie McNamara moving to left-back and Marc Twaddle pushing into the centre.
Motherwell continued to press and Ross Forbes' cross was a fraction too high for Coke to convert.
Carl Finnigan's first touch let him down after Arfield played him through but the striker almost atoned when he headed towards the far post following Jack Compton's corner, with Lasley clearing well as Enoch Showunmi tried to pounce.
Falkirk improved and Arfield and Darren Barr missed half-chances before their best chance of the half fell to Showunmi. Colin Healy played Compton's deep corner back across the face of goal but the former Leeds striker headed into the arms of the grounded Ruddy.
The game continued to pick up and Barr saved Falkirk with a last-ditch challenge on Forbes before Robert Olejnik saved headers from Forbes and Stephen Craigan.
Coke was sent off after a high and late tackle on Arfield, but the 10 men started the second half the better side and Olejnik saved two curling efforts from Forbes, before Murdoch survived a penalty claim when he nudged John Sutton as the striker attacked Keith Lasley's cross.
The substitute almost set up the opener when his low ball down the right found Arfield's run but the midfielder's attempted chip was clawed away by Ruddy's fingertips and Showunmi headed over Compton's cross from six yards seconds later.
Compton could have seen red too after a poor challenge on O'Brien, just moments after being booked for bringing down the Motherwell winger. Falkirk enjoyed a decent spell of possession but Jutkiewicz was inches away from Tom Hateley's cross after replacing Jamie Murphy.
The Bairns brought on attackers Mark Stewart and Pedro Moutinho but they still struggled to get in behind the visitors.
And they had an escape when Jutkiewicz was harshly flagged offside after O'Brien played him clean through following a poor goal kick, the whistle going before the striker slotted home.
Falkirk claimed for a goal in the 80th minute when Ruddy stumbled backwards as he caught Murdoch's deep cross underneath his crossbar. Barr got away with a clear penalty-box tug on Jutkiewicz's shirt before Olejnik held Craigan's header from a Hateley corner, and Reynolds raced back to take the ball off Moutinho after a break of the ball had allowed the substitute a run on goal.
Ruddy secured the point in the 87th minute when he acrobatically clawed the ball away from his top corner after Compton's driven cross had looped off Hateley's head.