The two-goal lead came under immediate threat as Ainsley Maitland-Niles tripped Pereyra with referee Martin Atkinson awarding Watford a penalty, with Deeney's effort saved by Cech this time around.
There was to be no late dramas for Wenger and his players to contend with as they now look forward to completing the job against Milan on Thursday and progressing into the last eight of the Europa League.
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The north London side can still qualify for Europe's premier club competition if Arsenal manage to win the Europa League just as Manchester United managed to do under Jose Mourinho last term.
We knew we're on a hard run at the moment, we had to look at ourselves and we had a big result in midweek in Milan and I felt we carried the determination into today.
The Hornets had chances to level in the first half, though none were as good as Deeney's opportunity from the spot after the break.
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"It was a convincing win". We had a disappointing period and our supporters suffered like we suffered. We have to get the fans on our side, and do absolutely everything to achieve it.
And Wenger piled on the praise for his goalkeeper and his "remarkable" achievement. We need some more games.
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With a massive 4-1 victory over Bournemouth, Spurs jump into third place in the Premier League with eight matches to play. Until tomorrow or after tomorrow he make the scan we can not say nothing more ", said Pochettino.
Watford benefitted from a penalty decision against the Gunners, and won the match with a last-minute Tom Cleverley goal, and Deeney revelled in the victory afterwards. But I like both situations, because in the first he showed a lot of humility and responsibility, and a record like that witnesses a lot of motivation and a huge degree of intelligence.
"Sometimes in the modern world it's not always easy because words are taken out of context".
And Wenger made a point of mentioning the incident after the match saying: "I wanted to win this game". "I believe that I like him as a player and when you get negative statements you have to deal with it".
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By that stage the action on the pitch had become nearly incidental with fans directing their anger at the club's chiefs. Again, everybody knows that you don't cross the line, you don't come on the pitch - that's the rules.