The visitors appeared on course to make it five wins in a row in all competitions thanks to Per Mertesacker's first-half goal - only to concede twice in a dramatic final 20 minutes at Vicarage Road.
Deeney equalised from the spot after Hector Bellerin was adjudged to have fouled Richarlison inside the area, a decision Arsene Wenger branded "scandalous" in his post-match interview .
However, Deeney - who also had a hand in his side's winner, challenging goalkeeper Petr Cech for a loose ball that eventually broke into the path of scorer Tom Cleverley - insisted Arsenal's failure to hang on had nothing to do with referee Neil Swarbrick's penalty call.
90 - Arsenal have lost to a 90+ minute winner in the Premier League for the first time since January 2012 (Bobby Zamora for Fulham). Stung.— OptaJoe (@OptaJoe) October 14, 2017
"I've heard Wenger's already blaming that [the penalty] as the reason they lost," Watford's club captain told BT Sport.
"Well, I'm not one to tell Mr. Wenger about himself, but the reason they lost wasn't due to one penalty.
"Having a bit of cojones, I think the word is. Whenever I play against Arsenal – and this is just a personal thing – I go up and think 'let me whack the first one, then we will see who wants it'.
"I came on today and jumped up with Mertesacker – I didn't even have to jump up actually – nodded it down and the crowd gets up. They all just backed off.
"For me, as a player, I just think happy days. That's my strength, so if you're going to let me do that against you, you're in for a tough afternoon."
Deeney also had harsh words for Arsenal midfielder Mesut Ozil, who was guilty of missing a one-on-one chance by shooting straight at Heurelho Gomes.
"[They didn't have] the ruthlessness," Deeney continued. "I don't think there’s many top Premier League teams that would have the chance that Ozil had and be so naive and just nonchalant with it.
"I'm old-school Arsenal. I loved watching Arsenal, my son's an Arsenal fan. [Ian Wright] doesn't miss those. Thierry [Henry] doesn't miss those. Even [Emmanuel] Petit, [Patrick] Vieira, [Paul Merson]… they're ruthless.
"He missed it and was like 'Ugh, we'll get another one'. Unfortunately, this is the Premier League now and everyone's catching up. You don't get another one."