Wales travel to Georgia in the Group D qualifier on Friday night and they will be without their talisman Bale, who has a calf injury, but Reading midfielder Edwards has tipped 17-year-old Woodburn to follow in the footsteps of the Real Madrid man.
"I've seen a lot of Ben in this campaign and he reminds me of what Gareth was like when he was younger," Edwards said.
"And if you see what Gareth was like in the Euros, the bigger the stage, the more he thrives. There aren't many players who can do that when the whole world is watching."
Wales currently sit in second place in their group and in order to guarantee qualification to next summer's tournament they must win both of their remaining qualifiers.
"From what I've seen, I'm pretty sure Ben is cut from the same cloth and he will relish the chance to show everyone he is the real deal."
Chris Coleman's side have not won a match without Bale since 2013, and with the former Spurs man set to miss the final qualifier against Republic of Ireland on Monday, Edwards says the rest of the squad needs to step up in his absence.
"I don't think any player could give more on the pitch but we maybe need to take more responsibility," Edwards added.
"Gareth has that ability to win a game on his own, producing that bit of magic. But we have other players who can do that as well, someone like Aaron (Ramsey), who obviously did that out in Serbia."