Neil Warnock's resignation in the wake of their Easter Monday defeat to Derby saw United travel under the caretaker pair of academy boss Neil Redfearn and former club captain Richard Naylor.
But neither could prevent the Whites from slipping to a fourth straight league defeat as they also extended their winless run on the road to 11 games.
Luke Varney cancelled out Johnnie Jackson's opener before substitute Obika stole it at the death as Charlton recorded their third win in five matches.
Green whipped in a dangerous ball as Leeds took the game to their opponents but Steve Morison failed to beat Ben Hamer who showed good handling.
The hosts were almost gifted the lead on 12 minutes when Whites defender Jason Pearce sliced a cross towards his own goal but Paddy Kenny did well to push the ball away under pressure.
Yann Kermorgant was next to test Kenny with an acrobatic scissor kick but again he held firm as the hosts began to find their feet in the final third.
Their best chance of the half fell to Dorian Dervite, whose low effort was beaten away by Kenny before Michael Morrison lifted the rebound over from close range.
Both sides looked better going forwards than defensively, and Leeds carved out another opportunity on the break as Morison played in Green but the Republic of Ireland international could only fire over the top.
The deadlock was broken immediately after the restart when Andy Hughes, a former cult hero at Elland Road, saw his effort diverted in by Jackson for his 10th goal of the season.
The decision paid dividends when the winger thundered in an equaliser nine minutes from time after Stephen Warnock's delivery was not cleared away.
Charlton protested there was a handball in the build up and were soon questioning referee Stuart Attwell again shortly after when Kermorgant went down inside the area under pressure from Whites skipper Lee Peltier.
Chris Solly saw his effort blocked as the hosts pushed forward for a late winner and they reaped the rewards in the sixth minute of added time when Obika headed home a Rhoys Wiggins cross.