Roy Hodgson's side had given themselves a two-goal lead to take to Perm, and apart from a major scare in the closing minutes they were comfortable.
Kushev powered home Georgi Peev's cross from the right, via the underside of the crossbar, and the home side passed up a late chance to win the tie on the away goals rule when substitute Nikolai Zhilyaev shot over the bar from eight yards.
Fulham were without USA international Clint Dempsey who was rested after taking a knock on his ankle in training, while strikers Andrew Johnson and Bobby Zamora, midfielder Danny Murphy and defender Paul Konchesky were also out injured.
The hosts dominated the first half but created little in the way of chances.
Fulham barely entered the final third and Erik Nevland cut a lonely figure alone up front.
The best chance of the half fell to Amkar's Jean Carlos in the 17th minute when Miklos Gaal crossed from the left and his deep delivery was played back into the danger area by Peev.
Jean Carlos shot from eight yards but Mark Schwarzer reacted excellently to scramble the ball away, and the same player's attempt to convert the loose ball failed as Fulham cleared.
The same combination that would later provide the game's only goal almost created an opener in the 50th minute when Peev's right-wing cross found Kushev in space by the penalty spot but the Bulgarian fired wide with the goal almost at his mercy.
Amkar broke at pace in the 72nd minute and had Fulham stretched, however John Pantsil managed to make a fine recovery to block Vitaly Grishin's shot.
A goalless draw looked inevitable but Amkar made a late breakthrough when Kushev's header flew past Schwarzer.
That meant Amkar, who snatched a late goal at Craven Cottage last week through Grishin, to keep alive their hopes for the second leg, needed just one more goal to qualify and knock out their visitors from London.
Their big chance arrived one minute into stoppage-time as they broke out of defence but Zhilyaev's shot was wayward and Fulham could breathe again.
Hodgson's troops could then have scored themselves when Diomansy Camara created a chance for substitute Seol Ki-Hyeon.
The midfielder took an unnecessary touch on the edge of the penalty area before placing his shot just wide, but the miss did not prove costly.