Unai Emery's side have struggled for form on their travels in all competitions but were gifted three points at Vicarage Road on Monday, moving them up to fourth in the table.
Foster's lapse in concentration saw Pierre-Emerick Aubameyang score his 18th league goal of the campaign in farcical fashion in the 11th minute, with Deeney then almost immediately dismissed for an elbow on Lucas Torreira.
FA Cup finalists Watford impressed despite being reduced to 10 men, full-back Adam Masina clipping the crossbar with a superb drive in the second half, but they succumbed to their first home defeat of the year.
Watford made the better start only to fall behind in self-inflicted circumstances.
Daryl Janmaat played a routine pass back to Foster but the Watford goalkeeper did not spot the advancing Aubameyang and his clearance crashed against the striker and into the net.
Within two minutes Watford were down to 10, Deeney sent off for catching Torreira, although Bernd Leno soon tipped Craig Cathcart's drive on to the post.
The Hornets continued to test Leno despite Arsenal's man advantage, with Etienne Capoue's floated free-kick turned away by the keeper.
Foster made a fine block to deny Henrikh Mkhitaryan after the restart, the Armenia international seeming certain to turn home Alex Iwobi's cross with Watford caught short at the back.
Masina went closest to earning the hosts what would have been a deserved draw, the defender letting fly with a rising drive that nicked the woodwork.
Arsenal failed to kill the game, Aubameyang missing a chance for his second, but they held on thanks to Ainsley Maitland-Niles' excellent block on Andre Gray to move within a point of third-placed rivals Tottenham.
What does it mean? Arsenal finding consistency
Emery's men had only beaten Huddersfield Town away from home in the league since November, but they finally seem to be finding their feet despite Deeney's red aiding matters.
Watford, meanwhile, will rightly wonder how the game may have played out differently had they competed on a level playing field. They did not look overawed at all.
Xhaka makes welcome return
With big games coming every few days, Emery was able to welcome back Switzerland international Granit Xhaka for his first club appearance in over a month. Xhaka is not the silkiest of midfielders but he adds much-needed steel and grit to Arsenal.
Deeney demeans the armband
Watford captain Deeney was incensed by Craig Pawson's decision to send him off but the referee, who consulted with his assistant before producing the red card, had little option.
Deeney memorably highlighted what he described as Arsenal's lack of "cojones" after the Hornets came from behind to beat the Gunners in October 2017, but he let his side down here.
Key Opta facts
- Since returning to the Premier League in 2015-16, Watford have picked up 15 red cards in the competition, more than any other side.
- Pierre-Emerick Aubameyang has been directly involved in three goals in his three Premeir League appeareances against Watford, scoring two and assisting one.
- Aubameyang netted his first goal away from Emirates Stadium since Boxing Day (vs Brighton), ending a run of seven games on the road in all competitions without scoring for Arsenal.
- None of the last 12 Premier League meetings between Watford and Arsenal have ended in a draw (10 wins for Arsenal, two for Watford); only two fixtures in the competiton have been played more often without ever sharing the spoils (Manchester United v Wigan (16), Aston Villa v Derby (14).
Arsenal go to Napoli seeking a Europa League semi-final place on Thursday before hosting Crystal Palace three days later in the Premier League. Already relegated Huddersfield welcome Watford on Saturday.