Martin Kushev headed a late winner as Amkar Perm won 1-0 but Fulham completed a 3-2 aggregate victory in Russia to secure their place in the Europa League group stages.
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, while Erik Nevland cut a lonely figure up front for Fulham.
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 towards the Brazilian striker who was thwarted excellently from eight yards by Cottagers keeper Mark Schwarzer.
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.
A goalless draw looked inevitable but Amkar made a late breakthrough when Kushev's header flew past Schwarzer.
Amkar's chance to snatch the tie on the away goals rule 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, but the midfielder took an unnecessary touch on the edge of the penalty area before placing his shot just wide.
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