And they got the goal they deserved - and a fair slice of luck - when Sidwell's effort deflected wickedly off Carew to nestle into the bottom corner.
The lead might have been added to but Young's free-kick hit the bar and Carew's first time shot was brilliantly beaten away by Vaniak.
And Villa almost paid for not taking their chances when Jarolim went past two defenders in the box and fired in a low shot but Guzan made another fine stop.
That should have been a warning to Villa, but it served only to inject confidence into the hosts.
A cross-shot from Krajcik fizzed past the post as Slavia raised the tempo and Guzan was again called into action to keep out Jarolim's fierce drive.
With Villa pegged back deep in their own half, their lead was looking more and more vulnerable.
But Necid headed wide and later drove a shot off target as the equaliser continued to prove elusive.
Some desperate defending from Villa just about saw them through - and it leaves them in a strong position after two matches.
Aston Villa boss Martin O'Neill: "We were really great in the first half. Maybe we could have scored more goals.
"I think we deserved to win. I believe now that seven points will definitely get us through to the play-off stage scheduled for next spring."
"I think Guzan is a great goalkeeper in the making and I'm delighted he got his work permit, and I think he's following in a long line of great American goalkeepers.
"I thought he was brilliant and all the things we felt we saw in him when he was in America, we're delighted they've come out."
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