City have a game in hand on top of their slender lead, but the Gunners are refusing to give up the fight in their hunt for a first league title in 19 years.
Newcastle's second home defeat of the season slows the Magpies' charge towards Champions League football next season as their lead over fifth-placed Liverpool remains three points.
Arsenal held an eight-point lead at the top of the table just a month ago but had won just one of their previous five league games to let the destiny of the title slip out of their hands.
Mikel Arteta's men, though, showcased their improvement over the last 12 months since defeat at St James' Park last season cost them a place in the top four.
"It is a big step for a young team like ours to come here and do the things we did. It shows we have come a long way," said Odegaard (24).
"Last year here was one of the toughest days of my career, to be honest. After dropping points against Man City, to win against Chelsea and then come here and win shows the mentality.
"We have to keep going and digging in and fight until the end. It is football, anything can happen and we need to be ready."
The visitors wobbled early on as Jacob Murphy smashed the inside of the post after just three minutes before the home side thought they had a penalty when Bruno Guimaraes' effort hit the arm of Jakub Kiwior.
However, the Polish defender was handed a reprieve by a VAR review as the ball appeared to come off his thigh first.
Having ridden their luck, Arsenal took full advantage to go in front with their first shot on goal as Odegaard took aim from long range and arrowed the ball low beyond Nick Pope's left hand on 14 minutes.
Martinelli magic
Pope then had to make a series of big saves just to keep Newcastle in a pulsating game before the break.
Three times the England goalkeeper stood up tall to block one-on-ones with Bukayo Saka, Gabriel Martinelli and Odegaard.
The unrelenting pace did not let up after the break as both sides hit the woodwork inside the first six minutes of the restart.
Alexander Isak's header came off the post before Aaron Ramsdale produced a remarkable save from Fabian Schar to keep Arsenal in front.
At the other end, Martinelli finally had Pope beaten only for the ball to come back off the crossbar.
Tempers started to flare on both sides of a feisty contest before another piece of brilliance from Martinelli produced the crucial second goal.
The Brazilian ran half the length of the field and then fired in a low cross that Schar could only turn into his own net.
While Arsenal still need favours from City in the run-in to achieve their goal, a top-four finish is still within Newcastle's reach.
Eddie Howe's men likely need just six points from their final four games to guarantee Champions League football for the first time in 20 years.
However, they could slip to fourth later on Sunday if Manchester United win at West Ham.
"We have four games remaining and need three results," said Howe. "Liverpool and other teams are coming. They are top level.
"We can only do what we can do and I believe in the character of the team to get it over the line."
