Despite losing Brophy in the 23rd minute, Killie battled valiantly to secure all three points courtesy of Nicke Kabamba's third goal of the season just before the hour mark.
Hamilton were forced to shuffle the pack due to Covid-19 protocols as Kyle Munro and Ross Callachan dropped out and Kilmarnock started brightly with the evergreen Burke prominent down the right flank.
The 36-year-old attempted to recreate his stunning free-kick against Hibernian earlier in the season but dragged it well wide in the seventh minute.
It was no surprise that the former Scotland international was instrumental in the opening goal in the 10th minute.
The visitors offered little as an attacking threat until Odoffin equalised in the 18th minute. David Templeton's pinpoint corner found the defender unmarked six yards out and after his initial header was parried by Danny Rogers, he slammed the ball into the roof of the net.
The equaliser seemed to boost the visitors' confidence and they almost went ahead minutes later but David Moyo fired straight at Rogers from inside the box.
An entertaining encounter was clouded by a controversial red card for Brophy in the 23rd minute.
The former Accies man stretched to block Scott McMann's clearance but caught the defender and Walsh brandished a straight red.
Kilmarnock looked the most threatening as half-time approached, Burke blazed wildly over the bar before Kabamba comically stood on the ball after being released clean through on goal.
Kabamba put Kilmarnock back in front in the 57th minute and Hamilton made a host of tactical and personnel changes as they chased an equaliser, but they will be disappointed with their inability to test Rogers.
Their best chance fell to Will Collar but his half-volley from the edge of the area drifted wide.
The hosts looked in complete control with captain Dicker a dominant presence, and Aaron Tshibola almost killed the game on the counter-attack but he was denied by Fulton with just over 10 minutes remaining.