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.

Five workers from Scottish Water dug a two metre (6ft) hole and spent 12 hours retrieving the rubbish dumped down a manhole at Milnafua near Alness.
Scottish Water said it believed someone had used the pipe as a fly tip.
Other items pulled out included a yellow cricket bat, a sandcastle mould and a paint pot.
Garry Kirkwood, Scottish Water's customer manager for sewer response, said: "It looks like someone has looked to use this manhole as some kind of tip.
"Some may see comedy about this incident but there is an underlying serious matter - sewers are not a dumping ground for household refuse or unwanted items.
"Blocked sewers cause serious problems to the network such as floods that affects homes, businesses and roads as well as the cost of having to get crews out to clear the blockages."
He added: "To have a crew of five to be tied up for 12 hours on this urgent clear-up job also means that other repair incidents had to be delayed."
It is not the first time Scottish Water workers have had to recover large objects from blocked sewers.
DuringI 2014, the company said its staff had pulled a large Winnie the Pooh teddy, a bike, a fax machine, jeans and a snake from Scottish sewers.
A goal from substitute Eamonn Brophy 17 minutes from the end earned Hamilton a point against 10-man Celtic.
[NEXT_CONCEPT]
A plastic rocking horse, a dining chair, a child's scooter and a football were found to be blocking a sewer pipe in the Highlands.