The Gunners were frustrated for most of the encounter but skipper Odegaard popped up to score the winner with 18 minutes remaining to extend his team's lead at the top of the standings to six points.
It was also a fifth consecutive league win for Arsenal since their 3-0 defeat by Brighton in September - still the only time they have dropped points so far this season.
The home side went close through Declan Rice's first-half header, saved by visiting goalkeeper Konstantinos Tzolakis, while Noni Madueke also went close from a tight angle.
Mikel Arteta's team also had a penalty shout turned down in the first period after Kai Havertz's shirt was pulled in the box.
The decision was not given on the field although the Video Assistant Referee told referee Farai Hallam to look at the pitchside monitor.
Hallam then informed the crowd Havertz had been fouled, but no penalty would be awarded as the forward had been in an offside position.
Nottingham Forest claimed a vital 3-1 victory over Manchester City to register their first home Premier League win of the season, with Liam Delap scoring twice in an impressive display at the City Ground.
Noah Okafor scored a decisive 79th-minute winner as Leeds United claimed a valuable 1-0 Premier League victory over Tottenham Hotspur at Elland Road.
Xabi Alonso has urged Cole Palmer to put England's latest international snub behind him and concentrate on producing his best football for Chelsea.
Unai Emery has confirmed he is discussing a new contract with Aston Villa, signalling his desire to continue building the club despite reported interest from Manchester United and other teams.
