Jelavic headed home just seconds after kick-off at the Gerhard Hanappi Stadium to secure a victory the Austrian side deserved for their dominance in the rest of the game.
Villa boss Martin O’Neill said: “We knew this was going to be exceptionally difficult – so will the return tie – but we are going to try to win.
“We are still well in the game and have a great match to look forward to next week.”
Christopher Drazan was a constant threat for Rapid and had one good shot cleared off the line by Curtis Davies while Brad Guzan saved another.
Villa’s best moments came in the second half but Davies headed over from one clear-cut opportunity and the visitors failed to seriously test opposing goalkeeper Helge Payer.
Guzan was given a rare start in the Villa goal in one of four changes made by manager Martin O’Neill following last weekend’s defeat by Wigan.
But the first job of the American was to pick the ball out of the net.
Only seconds had gone when the ball was worked out to the dangerous Drazan on the left and he whipped in a curling cross for Jelavic to head home.
Villa responded with Craig Gardner firing over while an Ashley Young cross was punched clear by Payer.
Villa made an encouraging start to the second half and the introduction of Gabriel Agbonlahor added pace. Shorey was given a shooting chance but fired into the side-netting.
Young created Villa’s best opportunity with a brilliant turn and cross just after the hour but Davies was unable to keep his header down.
Milner and Young combined at a free-kick but the winger’s shot was just off target.
Villa finished strongly but Milner could not make the most of Young’s cross and Sidwell shot straight at Payer.
RAPID VIENNA: Payer, Patocka, Dober, Katzer, Jovanovic, Heikkinen, Hoffmann, Drazan (Boskovic 83), Konrad (Trimmel 62), Pehlivan, Jelavic. ASTON VILLA: Guzan, Shorey, Davies (Lowry 81), Cuellar, Beye, Ashley Young, Sidwell, Reo-Coker, Gardner (Agbonlahor 55), Milner, Heskey. Referee: A Tudor (Romania)