A needless foul from Dougie Imrie on Leigh Griffiths brought Celtic a penalty, and the striker stepped up to slot home his 33rd goal of the season.
Dedryck Boyata was sent off for a last-man tackle on Accies' Carlton Morris five minutes before the break.
Michael McGovern saved a second Griffiths spot-kick, and Brophy's powerful strike rescued Accies a point.
They move above Motherwell into ninth in the Premiership table, two points above the relegation play-off place currently occupied by Kilmarnock.
Celtic extend their cushion at the top of the table to seven points, but Aberdeen can take advantage of the champions' slip-up by closing the gap back to four points when they host St Johnstone on Saturday.
Hamilton were a world away from January's humiliating 8-1 thrashing at the hands of the same opponents, and very nearly took the lead through Ali Crawford.
Scott Brown lost out to Morris who put in Crawford though on goal, but he shot over the bar when he really should have scored.
Accies had settled into the game after an dominant opening period from the visitors, but fell behind due to a moment of madness from Imrie.
Griffiths was in possession but in an unthreatening position and only Imrie will know why he sent the Celtic striker crashing just inside the area.
The striker took the penalty himself and fired the ball right down the middle.
It was a crazy moment from Imrie, a point Mikey Devlin was presumably making as he squared up to his team-mate in the wake of incident.
But suddenly the game was alight.
Boyata was sent off for a tackle on Morris. He tried to play the Accies striker offside but mistimed his movement and the forward was in on goal.
The lunging challenge ended a clear goalscoring opportunity just outside the box, but the Celtic bench - and their support in a crowd of 5,017 - clearly felt the tackle was legitimate and the defender had touched enough of the ball.
Manager Ronny Delia immediately reshaped his 10 men, replacing the returning Kris Commons with Charlie Mulgrew.
The flames continued to burn in the second half. Gary Mackay-Steven was through one-on-one with McGovern but could not finish, and Griffiths saw a clever effort fly just past the post.
Then came another moment of controversy and another penalty. Ziggy Gordon was spotted wrestling with Mikael Lustig at a corner and referee Craig Thomson pointed to the spot.
Again Griffiths stepped up, but this time McGovern produced a fine, diving save to keep the outcome in the balance.
It was a costly miss because minutes later home boss Martin Canning introduced Brophy - criticised by some after celebrating his goal in the recent thumping at Parkhead - and the substitute had the desired effect.
Celtic failed to deal with a corner and when the ball dropped in the box to Brophy he smashed a finish high into the net to ensure honours were even at the end of a pulsating match.
A goal from substitute Eamonn Brophy 17 minutes from the end earned Hamilton a point against 10-man Celtic.