Arsenal now sit in fifth place in the table, while Watford are sixth after extending their winless run in the top flight to three matches.
Both teams went into the fixture with similar levels of confidence having enjoyed a strong start to the campaign, and it came as little surprise when the two sides sprayed the ball about with confidence in the early stages.
Arsenal created a chance of their own shortly afterwards as Alexandre Lacazette dispossessed Cathcart before running clear on goal, but the Frenchman could only dink his one-on-one effort wide of Ben Foster's goal.
The two clubs continued to show intent in the final third, but chances were few and far between as the defences remained on top for much of the first half.
Arsenal had another opening towards the end of the first 45 minutes with Granit Xhaka picking up the ball on the edge of the penalty area, but the Swiss's long-range strike was tipped away from goal by Foster.
Both teams went onto the offensive after the restart and Bernd Leno - on as a substitute due to a hamstring injury to Petr Cech - pulled off a brilliant save down to his left to prevent Troy Deeney's volley from finding the bottom corner.
Arsenal responded with a rapid counter-attack which ended with Pierre-Emerick Aubameyang crossing for Lacazette who was waiting by the penalty spot, but the Frenchman headed wide of the far post from 12 yards.
The Gunners dominated the next stage of the game but by the start of the final quarter, Watford were beginning to trouble their hosts with Gray being denied by a smart block from Leno.
Substitute Isaac Success should have opened the scoring shortly afterwards when breaking clear of the Arsenal defence, but the forward's shot from 10 yards clipped the far post and went wide.
The game appeared to be heading towards a goalless draw but after the contest had entered the final 10 minutes, Arsenal struck twice in the space of three minutes.
The first came via the boot of Watford defender Cathcart, with the Northern Irishman diverting the ball inside the near post after Lacazette had failed to get on the end of Alex Iwobi's low cross.
The second arrived in similar circumstances with Lacazette breaking down the right flank before crossing for Ozil, who sent a confident first-time strike to the left of Foster from 10 yards out.
Watford attempted to find the consolation goal that their play probably deserved, but the Hornets were left to watch Arsenal see out a win which takes the Gunners above them in the top-flight standings.
ARSENAL (4-2-3-1): Cech (Leno 45'); Bellerin, Mustafi, Holding, Monreal; Torreira, Xhaka; Ozil, Ramsey (Iwobi 63'), Aubameyang (Welbeck 77'); Lacazette
WATFORD (4-4-2): Foster; Navarro (Femenia 84'), Cathcart, Kabasele, Holebas; Pereyra, Capoue, Doucoure, Hughes; Gray (Success 72'), Deeney