Homegrown hero Ryan Sessegnon's composed finish after Raul Jimenez's header was palmed into his path settled a tie of few clear-cut chances at Craven Cottage.
Villa had hoped to further cement their Champions League qualification hopes and leapfrog Man Utd into third - but lacked any kind of cutting edge and instead could be overtaken by fifth-place Liverpool later on Saturday.
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Ollie Watkins and Morgan Rogers both put first-half efforts narrowly wide, but not until Tammy Abraham's glaring 84th-minute miss did they fashion another chance of real quality, despite controlling possession throughout the second period.
The visitors had been happy to cede possession and territory to Fulham at first and frustrated their hosts well, but lacked the bite or fluidity to hurt them on the break until Rogers and Watkins' efforts suggested things may be beginning to click.
Instead, Fulham finally found their own finishing boots to score for the first time since March 21 after Sander Berge was allowed to stride forward and cross from the right flank, with Jimenez's diving header saved by Emiliano Martinez and Sessegnon alive to guide home the rebound.
Villa emerged a different side, playing 10 yards further up the pitch but still without the clinical touch required, and were almost punished again when Timothy Castagne nodded home Sasa Lukic's corner, only for Sander Berge to be penalised for a foul on Martinez.
It left the door open for Villa to rescue a deserved point and they should have levelled when substitute Abraham, the hero a week earlier against Sunderland, danced beyond Andersen before shanking an effort over with the goal at his mercy - to leave Fulham dreaming of a repeat of their run to the Europa League final 16 years ago.
More to follow...
