Playing three at the back in the first half, the home side struggled to retain possession and eventually fell behind when Per Mertesacker headed home Granit Xhaka's corner six minutes before the break.
Having told his players at the interval to be fearless in their approach, Silva watched on proudly as Watford fought back to take all three points.
Substitute Troy Deeney converted a 71st-minute penalty before Tom Cleverley smashed in the winner in stoppage time, in the process sending his side above Arsenal and into the top four of the Premier League.
"The first half we were shy in some moments," Silva told BBC Sport.
"I think we played for 10 or 15 minutes like I wanted us to. We started well but then we gave away the ball too easily – we started to play without confidence.
"I didn't like the last 10 minutes of the first half. We spoke well in half-time, I showed our players the plan, what they needed to do.
"We needed to play without being scared. Even against big teams like Arsenal, who have a lot of quality, you need a lot of quality as well.
"I think our second half was then really good."
However, Silva insists they should not get too carried away with their fine start to the season — even though they are now sitting in the Champions League places.
"It's good for everybody. Our fans were fantastic again today," he added.
"We got three important points. A lot of games to play yet, we will stay in a good moment — and to do that we will have to work really hard."