Goalkeeper Foster gifted the Gunners an opening goal after 10 minutes when he took too long on the ball after collecting a back pass, allowing Pierre-Emerick Aubameyang to charge down his clearance and deflect the ball in.
Just a minute later, Watford captain Deeney was sent off for throwing his forearm into the face of Arsenal midfielder Lucas Torreira in a late challenge.
Despite playing with 10 men for 80 minutes, Watford offered the far greater threat, hitting the bar through Adam Masina’s powerful drive from distance in the second half, and forcing Arsenal goalkeeper Bernd Leno into a number of excellent saves.
The German denied Craig Cathcart well with his feet shortly after Deeney’s red card – and later in the first half dived at full stretch to tip away Etienne Capoue’s goalbound free-kick.
Ainsley Maitland-Niles also produced a crucial block with 10 minutes remaining when he slid in to block Andre Gray’s shot after the Watford striker had rounded Leno.
The visitors’ only real chance came midway through the second half but Foster brilliantly spread himself to block Henrikh Mkhitaryan’s volley from close range.
The Gunners were well below their best but the rare away win could prove crucial in the race to qualify for next season’s Champions League.
They climb two places to fourth in the table, two points clear of Manchester United and ahead of Chelsea on goal difference.