The result lifts Arsenal to 68 points — one ahead of Jurgen Klopp’s Liverpool, who host bottom side Sheffield United on Thursday.
Elsewhere on Wednesday, Phil Foden scored a hat-trick as defending champions Manchester City crushed fourth-placed Aston Villa 4-1 to move level on points with Liverpool.
“We want to be at the top — winning our games is the only thing we can do,” Arsenal manager Mikel Arteta told the BBC.
“We managed to rotate and freshen the team up a bit,” he added. “Those who came in did really, really well.”
Arteta’s men, with top-scorer Bukayo Saka absent, dominated possession in the opening stages at the Emirates but created few clear-cut openings in front of an expectant crowd.
But they broke the deadlock in the 24th minute, courtesy of a sweet left-footed strike by Odegaard.
Emile Smith Rowe, who has been a peripheral figure this season, dispossessed the dawdling Pelly Ruddock Mpanzu inside the visitors’ half.
He fed Odegaard, who played a one-two with Kai Havertz before stroking the ball home for his 10th goal of the season.
Smith Rowe impact
Arsenal doubled their lead in the 44th minute thanks to an own goal from Luton defender Daiki Hashioka, who turned into his own net following more good work down the left by Smith Rowe.
The home side, playing in second gear, had just three shots on target in the first half but were well worth their lead.
The bulk of the second half was flat, although Luton came into the game more, probing for an opening to haul themselves back into the contest.
Arteta brought on midfielder Declan Rice and forward Eddie Nketiah, introducing Gabriel Martinelli with 15 minutes to play.
Arsenal substitute Takehiro Tomiyasu went close, bending one just wide, and Nketiah forced a save from the scrambling Thomas Kaminski in the closing minutes.
They were unable to score a third but saw the game out with few alarms.
The Gunners, who have not been crowned English champions since 2004, set the pace for the bulk of last season before flagging as the finishing line approached.
But, adding steel to their style, they have now won nine of their past 10 league games, with the only blip in that sequence a goalless draw against City on Sunday.
Arsenal, who have conceded just four goals in the league in 2024, travel to Brighton on Saturday before hosting Villa on April 14.
In between those matches they welcome Bayern Munich in the first leg of their Champions League quarter-final.
Defeat for depleted Luton leaves Rob Edwards’ side third from bottom of the English top flight, three points from safety, and facing an immediate return to the Championship.
Brentford drew 0-0 with Brighton to edge further clear of the drop zone.
© Agence France-Presse
Photo: Twitter @Gooner_Eurychrus