Without the England center back manning the back line, Everton's porous defense was lucky to concede only two goals to Arsenal in the first half.
Stones' second half introduction plugged the gap, but the 2-0 result was already in the cards. Arsenal's title chances remain a slim mathematical possibility with eight Premier League matches left, as they sit 11 points behind first-place Leicester City who won later in the day.
Arsenal won for just the second time in nine matches, withstanding a horrible recent run that featured home and away Champions League losses to Barcelona, a home Premier League loss to Swansea, an FA Cup home loss to Watford, and a preceding FA Cup home draw to Hull City.
"I thought the performance was top class," said Arsenal manager Arsene Wenger to BT Sport in post-match commentary, adding a subtle message to disgruntled fans who displayed anti-Wenger signs and lingered on well after the final whistle to voice their displeasure.
"We want everybody to be behind the team to give us a chance. We are ready for a battle, for a fight. Let's continue to produce the quality of the performance that we produced today," he said.
Danny Welbeck continued his fine form on return from injury, scoring his fourth goal in nine matches. It took a 12 pass build-up for the England international to score on a breakaway touch pass from Alexis Sanchez.
"It's been a difficult couple of weeks, so it was a good thing today to get back to our winning ways, and it's something we can build on from here," Welbeck told BT Sport. "To be able to get the early goal, it was good for us, and it set the standard for the rest of the game."
Nineteen-year-old winger Alex Iwobi scored his first goal for the Gunners in his debut as a starter. The Nigerian took two running touches from a 30-yard Hector Bellerin pass before drilling the ball past goalkeeper Joel Robles, who had recently won his first-team place over Tim Howard.
"I'm happy that the team won; it's a positive that we can take to the next game," said Iwobi, who seemed to relish his partnership with Welbeck on the front line. "I'm enjoying training and playing (with Welbeck), he gives me confidence and advice."
Iwobi appears to have leapfrogged longtime regular Theo Walcott in the attacking right-winger's position. Joel Campbell and injured Alex Oxlade-Chamberlain have also made effective appearances in that spot this season, leaving questions as to who would earn starting duties if all four are to return next season.
The final score did not reflect Arsenal's domination on the match, as Sanchez was denied a penalty after going down in the box in the 34th minute. The Chilean winger was clipped by Muhamed Besic but cast doubt on the foul by staying on his feet for just a moment too long before hitting the deck.
Retired referee Howard Webb, now an analyst for BT Sport, acknowledged that the decision should have gone to Sanchez.
An already depleted Arsenal side suffered another injury, as goalkeeper David Ospina withstood a lumbering knock by Romelu Lukaku inside the six-yard box. Ospina, who walked off limping after the final whistle, shook off the collision, sparing a debut for newcomer Matt Macey.
Arsenal's crowded injury room includes first-team regulars Petr Cech, Aaron Ramsey and Santi Carzola -- all due to return in two to three weeks, as well as Oxlade-Chamberlain who is out for another five weeks, according to arsenalreport.com.
Arsenal need as much manpower as they can get to stave off West Ham and Manchester United in their battle to maintain a top four finish for the 20th consecutive season -- all of them under the helm of Arsene Wenger.
Meanwhile, Everton have now beaten Arsenal just once in their last 18 meetings.
Ironically, the Toffees new 49.9% stakeholder, billionaire investor Farhad Moshiri, sold his Arsenal shares to take his seat at the director's box in Goodison Park. One wonders what mixed emotions ran through Moshiri's head as he watched his former club outclass his new squad.
"This is a setback. We have never performed in this manner," said Everton manager Roberto Martinez, who referenced the suspended Gareth Barry as a hole in the lineup. "There is nobody to blame apart from ourselves."
In Saturday's other Premier League matches, Leicester City maintained an eight point lead at the top of the table with a 1-0 win at Crystal Palace behind Riyad Mahrez's 16th league goal of the season.
Chelsea came from behind against West Ham twice to draw 2-2 at home, buoyed by two Cesc Fabregas goals. The Spaniard scored at the end of each half, first on a free kick and then on a late penalty after Ruben Loftus-Cheek was fouled by Michail Antonio in the box. The Hammers, who are in the midst of a historic season, missed a chance to move past Manchester City into fourth place.
Stoke beat Watford 2-1 away at Vicarage Road, while Norwich City beat West Bromwich Albion 1-0 away at The Hawthorns.
The day's late match featured two bottom five sides, with Swansea delivering Aston Villa's 21st loss of the year in a 1-0 win at home. Villa, who reached the FA Cup final last year, have scored just 22 goals in 31 matches this season.