In an exciting match played in front of over 38,700 spectators, it was Newcastle who took the early lead by scoring with their first attack in the sixth minute.
New signing Anthony Elanga made a fantastic start in the black and white by slotting the ball into the bottom right corner after connecting with a cross from Sandro Tonali.
However Arsenal overcame a shaky beginning, regaining their rhythm and control of the match.
Mikel Merino equalized for the Gunners with a precise shot in the 33rd minute following a flick from Kai Havertz.
They then took the lead two minutes later when Alex Murphy accidentally directed the ball into his own net.
The Magpies nearly conceded a third shortly after the restart in the 57th minute, but goalkeeper Nick Pope was instrumental in saving a bicycle kick attempt from Bukayo Saka.
Out of nowhere, substitute Jacob Murphy equalized for Newcastle just a minute later with a shot from outside the box that slipped past David Raya at his near post.
With the match appearing to be heading for a draw, it was 15-year-old Max Dowman's penetrating run into the box, where he was fouled by Joelinton, that turned the tide.
Odegaard then confidently converted the penalty to secure Arsenal's second win, following their earlier 1-0 victory over AC Milan.