It dealt a psychological blow to second-place Manchester City, who have headed into Saturday's late game at West Ham trailing Arsenal by 10 points.
A breathtaking finish at Emirates Stadium saw Dowman, an English soccer prodigy who is still at school, dribble around an Everton player inside his own half and race clear to tap into an empty net.
Everton goalkeeper Jordan Pickford was stranded upfield having gone up for a corner.
It was Dowman's first goal in his third Premier League appearance and he became the competition's youngest ever scorer.
The Gunners toiled before Gyokeres' late intervention, which came when a cross by Dowman from the right was missed by Everton goalkeeper Jordan Pickford, struck the midriff of Arsenal substitute Piero Hincapie and bounced across the goalmouth. Gyokeres applied the finishing touch from close range.
"He stays so calm and without fear every time he gets the ball," Gyokeres said of Dowman.
"He takes the right decision most of the time as well and with the goal, he definitely took the right one."
Across London, Chelsea lost 1-0 to Newcastle thanks to an 18th-minute goal by Anthony Gordon.
Chelsea stayed in fifth place but could be overtaken by Liverpool, who host struggling Tottenham on Sunday.
A 0-0 home draw with Bournemouth on Saturday left next-to-bottom Burnley eight points from safety with just eight games remaining this season and facing an immediate return to the Championship having won just four of their 30 league games.
Sunderland, another promoted team, appear to be safe from relegation but are limping toward the end of the season after a third straight home loss – this time to Brighton 1-0.
The only goal was a bizarre one, with Yankuba Minteh's mis-hit cross from the byline somehow squeezing in at the near post in the 58th minute at the Stadium of Light.
Sunderland hadn't lost at home until a defeat to Liverpool on February 11 but since then have lost to Fulham and now Brighton.
