The Group C clash at the Al Sadd Stadium had appeared to be headed for a goalless draw when the Ivory Coast striker, who had come off the bench in the second half, combined with fellow substitute Mohammed Razak before lashing the ball into the net to give the Qatari side their second home victory of the tournament.
It was a reversal of fortune for Lekhwiya, who had lost 2-1 in Iran two weeks ago when Bruno Correa headed in a last-minute winner for the hosts.
Wednesday's defeat was the first suffered in this year's competition by Sepahan, who could have wrapped up their place in the round of 16 if results had gone their way on Wednesday. However, the Esfahan club still remained a point ahead of Lekhwiya in the group with seven points.
There were few efforts of note in a dour first half. The best opportunity fell to Lekhwiya's Korean midfielder Nam Tae-hee, who sidestepped a challenge by Omid Ebrahimi just outside the Sepahan box before he launched a fierce shot that goalkeeper Rahman Ahmadi had to punch clear.
Algeria international defender Madjid Bougherra went close to unlocking the Sepahan defence midway through the second half with a marauding run that took him past six defenders before a last-ditch tackle saw the ball ricochet off him and into the grateful arms of Ahmadi.
Nam then let fly with a free-kick from near the right touchline that was initially misjudged by Rahmadi, who managed to get just enough of a touch on it to turn it onto the crossbar.
Sepahan's best opportunity of the half fell to Correa with five minutes left when the Brazilian was found in space on the left of the Lekhwiya box. However, he failed to add to his tally of three goals as goalkeeper Baba Malick came quickly off his line to snuff out the danger.
The match looked destined for a stalemate before Kone struck as the match entered stoppage time. The Ivorian played a neat one-two with Razak on the edge of the box before he thumped an unstoppable right-footed effort past Ahmadi.
Lekhwiya could even have made it two right at the end as Kone laid on a pass for an unmarked Razak, who fired well over the crossbar.
Lekhwiya assistant coach Jean Lang felt that the depth of his team's squad was crucial as they were able to bring on a pair of good impact players in Kone and Razak.
"We play as a team and having a strong bench is important especially when you get to 'magic time' so we're lucky to have a good bench," said the Frenchman.
"We won today against a strong team because Sepahan have more experience than us. This is our first season in the AFC Champions League so six points is good for us and we will try to continue to progress towards the second round."
Despite seeing his side overtaken in the group by Al Ahli, who moved into first place on goal difference after their 3-1 win over Al Nasr, Sepahan coach Zlatko Kranjcar remained confident of their chances of advancing to the last 16.
"I still believe that we will go through to the second round," said the Croatian.
"Lekhwiya are a very strong team and they have good technical players. I thought that we played very well tonight but this is football and anything can happen. It was a disappointing finish but we still have a chance to go to the second round if we do well in our remaining matches."