There was a concern for Saka as he limped off the pitch with an injury later in the second half.
The result leaves Newcastle winless after 13 league games and rooted to the bottom of the table.
"I am happy with the points after losing last weekend. You have to win straightaway. Overall, I'm pleased," Arteta said in his postmatch interview.
"The word was we had to be patient. We had to play with urgency and rhythm. That's what we did in the second half. We found spaces and that's how we scored the goals."
Newcastle keeper Martin Dubravka, recalled for his first appearance of the season in place of Karl Darlow after recovering from injury, produced an acrobatic save to keep out Martin Odegaard's free kick.
When Arsenal finally cracked Newcastle's defence after 55 minutes, it was no surprise Saka and Smith Rowe were involved.
The latter combined well with Tavares who slipped a pass in for Saka to fire left-footed past Dubravka from a narrow angle.
Martinelli, who had just replaced the injured Saka, was picked out by a flighted pass from Takehiro Tomiyasu and doubled Arsenal's lead after 66 minutes with a smart first-time finish. It was the youngster's second touch after coming on.
The visitors attempted to make a comeback but remain without a win as Eddie Howe watched on for the first time in the dugout after recovering from COVID-19.
"I thought it was a good performance, especially the start. We were good for the first half an hour," he said.
"The pleasing thing today is how we started and finished the game. We created chances."