Per Mertesacker had headed Arsenal ahead with his first league goal in 1400 days, only for a contentious Troy Deeney penalty to draw Watford level before Cleverley's late heroics sealed a memorable 2-1 victory.
"I would say it was not a penalty".
Marco Silva, Watford's manager, suggested that he could understand Arsenal's view but also said that Swarbrick had grounds to point to the spot.
The set-piece effort came after Arsenal had struggled to break down a Watford side that mirrored its opponents in starting with a three-man back line, too often settling for long-range attempts or throwing in hopeful crosses.
"It's important, I didn't see a dive or a simulation".
With Alexandre Lacazette becoming more accustomed to life in the Premier League, and full-back Sead Kolasinac proving to be one of the signings of the window in his free transfer from Schalke, Saturday evening's game should prove to be a tightly-contested match.
The opener came soon after as Mertesacker rose to head home Granit Xhaka's corner from six yards out, marking his first league start in nearly 18 months with a rare goal.
Moments later came the penalty drama, and then in injury time when Andre Carrillo's shot was blocked Cleverley lashed in a dramatic victor.
However Arsenal boss Arsene Wenger recently stated that Wilshere is getting close to returning to the first team when he said: "I hope he goes to the World Cup".
The midfielder came off the bench for the injured Danny Welbeck in the second half before wasting a golden opportunity to fire the Gunners into a 2-0 lead after failing to score with just Heurelho Gomes to beat.
"After that, at 1-1, there was no need to panic".
Nevertheless, Silva said: "I don't change what I said at the start of the season".