The match in Belgrade was suspended in the 41st minute when a drone carrying a flag with Albanian symbols flew over the pitch.
Serbia defender Stefan Mitrovic leapt to pull the flag down, but a melee ensued on the pitch between the two sets of players and a number of supporters entered the field of play as the situation escalated.
Referee Martin Atkinson ordered the teams to leave the field, but the Albanian players were then targeted by pitch invaders and pelted with missiles from the stands as they raced towards the tunnel.
Cikalleshi, who was a substitute for the Group I match, revealed the fear he and his team-mates had for their own safety.
"It was bad, we feared for our lives," the midfielder told Goal.
"In the beginning, we didn't want Serbian players to rip apart our flag which flew over the pitch. That's how it all started.
"We didn't feel safe. We didn't feel safe until we reached the tunnel and the locker room. Four Albanian players were injured in a brawl and we didn't want to continue the match after that.
"There were plans to keep on playing, but we didn't want to. Like all players on the bench, I ran onto the pitch but I wasn't punched, nor did I punch anyone.
"Right after the match, we were escorted to the airport by the police and flew to Tirana [the capital of Albania]."
Cikalleshi commended the actions of Serbia's players - and Branislav Ivanovic in particular - when the violence broke out.
"Serbian players wanted to protect us as fans entered the pitch. Branislav Ivanovic was there in particular to calm it down," he added.
"Even before the match, regarding everything that was surrounding it, he came up to us and said: 'Hey guys, it's only a football match'.
"But Mitrovic, who grabbed the Albania flag, surprised us all with that."
UEFA have confirmed they will open disciplinary cases against both nations due to the violence in the Serbian capital.
