Watford boss Quique Sanchez Flores, back in the dugout for a second spell at the club, saw his side start the game strongly but fall behind to two strikes from Pierre-Emerick Aubameyang before half-time.
The Gabon international turned and smashed home the opener on 21 minutes after good work from Sead Kolasinac and tapped home the second from a low cross by Ainsley Maitland-Niles.
However, Arsenal gifted Watford a goal back after the restart. Gerard Deulofeu intercepted a poor pass out of defence by Sokratis Papastathopoulos, and the ball broke to Tom Cleverley to rifle home a low shot.
The hosts went in search of an equaliser and levelled from the penalty spot on 81 minutes after David Luiz was penalised for a foul on Roberto Pereyra.
The Argentine stepped up to take the kick and sent goalkeeper Bernd Leno the wrong way to rescue a deserved point for the hosts.
Sanchez Flores made three changes to the Watford team, bringing in Jose Holebas, Etienne Capoue and Deulofeu. Arsenal were without Alexandre Lacazette due to injury but handed Ozil a first outing of the season.
TalkSport's Ian Abrahams shared the two teams:
Watford looked threatening in the opening stages on the counter-attack, with Deulofeu in particular causing the Gunners problems. The winger forced Leno into a save with a curling shot on six minutes, and the goalkeeper also had to be alert to tip a fierce effort from Cleverley over the crossbar.
Deulofeu set up Holebas for another shot on goal after wriggling past Maitland-Niles, but the 35-year-old could only blast an effort wastefully wide.
Arsenal made Watford pay for failing to take their chances by scoring with their first effort on goal. Dani Ceballos dispossessed Will Hughes just inside the Arsenal half, the ball breaking to Kolasinac to stride forward and find Aubameyang. The striker took a touch to control the ball and then turned and rifled a low shot past Ben Foster.
Aubameyang added his second 10 minutes later from a well-worked move by Arsenal. Ozil showed his quality with a clever pass to send Maitland-Niles into the box, and he squared to hand Aubameyang a simple tap-in.
Opta noted Arsenal's intricate build-up play:
However, it was a different story entirely after the break. The Gunners offered Watford a lifeline just minutes after the restart when trying to play out from the back. Deulofeu intercepted Papastathopoulos's intended pass to Matteo Guendouzi, the ball breaking to Cleverley to slam past Leno.
Arsenal's defending came in for criticism after the goal:
Cleverley's strike saw the momentum of the game switch in favour of the hosts. The Hornets pressed hard for an equaliser and were rewarded when Luiz conceded a penalty for a foul on Pereyra.
The Argentine sent Leno the wrong way with his spot-kick to level the match with less than 10 minutes of normal time remaining.
The host continued to attack in the closing stages and had chances to win it through Deulofeu and Abdoulaye Doucoure.
Sky Sports Statto highlighted Watford's dominance:
Sanchez Flores' side could not quite find an equaliser but deserve credit for their fightback. The Hornets highlighted Arsenal's familiar defensive frailties, and the result will bring fresh scrutiny on Unai Emery.
Watford's next fixture is against Manchester City at the Etihad Stadium on Saturday. Arsenal are at Eintracht Frankfurt in the UEFA Europa League on Thursday and then host Aston Villa in the Premier League on Sunday, September 22.