Spurs, reduced to 10-men for almost an hour, survived a late Villa fightback when Marc Albrighton scored.
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He made eight changes to the team, with Ashley Young, Stewart Downing and Marc Albrighton named as substitutes.
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Marc Albrighton slotted in from close range as he added to Villa's lead.
The visitors were rewarded for a strong start when Emile Heskey robbed Sebastien Bassong and squared for Marc Albrighton to slide in and convert.
Fans chanted: "You don't know what you're doing" when Houllier withdrew Marc Albrighton on 61 minutes and at the final whistle they chanted "we want Houllier out".
Marc Albrighton's cross drifted beyond Spurs goalkeeper Heurelho Gomes with eight minutes left to set up the prospect of a grandstand finish, but for all Villa's effort they could not snatch an unlikely equaliser.
Goals in each half from Stewart Downing and the fit-again Emile Heskey, both from Marc Albrighton crosses, ensured that Paul Scharner's late header counted for nothing as the impressive visitors pushed the hosts hard at Villa Park.
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Though Everton dominated possession, the fine link-up play down the left between Pienaar and Baines asked real questions of Villa right-winger Marc Albrighton - and though he was perhaps naive defensively, the 20-year-old showed real promise going forward.
It could have been even better for the home side but Sturridge headed wide from a late cross by fellow substitute Marc Albrighton and England will take a slender lead into the second leg in Botosani on Tuesday.
Marc-Antoine Fortune smashed a shot just wide soon after but seconds later Villa had scored thanks to Albrighton's magic on the right wing, where his instinctive trickery - nut-megging Marek Cech - allowed the young Englishman to get to the bye-line.
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