John Carew, who has been in fine pre-season form, opened the scoring after just seven minutes as Odense keeper Onysko could only parry an Ashley Young effort.
Odense equalised on 25 minutes as Villa debutant Steve Sidwell put the ball into his own net as the home side grew in confidence.
Martin Laursen thought he had claimed the win for Villa after he headed with 20 minutes to go but the away side were denied by an Anders Christianson leveller.
A draw was a fair result and gives the Claret and Blue a great chance to progress with the second leg at Villa Park next Saturday.
Villa will be very confident after an opening 20 minute spell of dominance when they could have found themselves three goals to the good.
After Carew's early goal, Young and Gabby Agbonlahor had efforts which they should have done better with.
Martin Laursen also tried his luck as Villa looked to kill the tie off quickly.
However, Odense were not disheartened and always knew that their opponents would become weary as the game went on.
And, the Danish side were fortunate as Sidwell deflected an Atle Atland shot into his own net.
They then began to threaten the Villa goal more frequently and after the break Stuart Taylor was forced into a fine save to keep his side level.
Craig Gardner then cleared off the line as Henrik Hansen forced the ball goalwards.
As his team mates continued to tire, Villa winger Young remained dangerous and he set up Sidwell who shot wide from the edge of the area.
Then came a goal which epitomised the Villa side of last season.
The effervescent Young swung in a trademark delivery for the towering Martin Laursen to head home.
It was a great moment for last year's Player of the Year Laursen, who opened his account for this campaign in his homeland.
The goal gave Villa a new lease of life and Young, Carew and Laursen again went very close before Christianson scored with just seconds to go.