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Adelaide pressed before the break and Goodwin weaved his way past two defenders before drilling an effort at goal, which was not held by Birighitti on the first attempt, but the keeper reacted well to scoop up the ball.
In the second half, Birgihitti continued to defy the home side as he made another fine stop to tip James Jeggo's powerful strike from the edge of the box past the post.
United continued to increase the pressure on the Jets' goal and Jeggo raided into the box to fire at goal, but Birighitti tipped his strike onto the bar and over.
The home side made one late surge for a winner, but Goodwin saw his close-range effort denied by Birighitti after he latched on to Sergio Cirio's cross into the box.
Adelaide remain bottom of the table following the draw, while Newcastle are still in fifth place.