Romelu Lukaku put Everton ahead after 17 minutes but lacklustre defending enabled Stefano Okaka to equalise before half-time.
Poor marking again proved costly as Watford scored twice from set-pieces in the space of five second-half minutes, Sebastian Prodl and Okaka again on target.
Lukaku pulled one back in the closing stages but Everton were unable to avoid a fifth match without a win, and Koeman was furious with how his team performed.
"The start was good. We were the dominant team at 1-0 up … but after that the problems started," Koeman told Sky Sports.
"We made some bad mistakes in our ball possession and had a lack of communication. In the second half the set plays killed us today.
"The team is too passive, it's not proactive, it's really reactive … that's maybe fear at the moment. They're still afraid at 1-0 up, [we] need that step of confidence. Until 1-1 there was one team on the pitch.
"Of course I'm worried as a manager at the moment because we've had a bad run of results. I see a lot of problems. We're too reactive and not taking the initiative in the game.
"I know it's maybe a lack of confidence, but if you go 1-0 up you need more belief in the team, not going back and being nervous. That's the big problem.
"I'm very happy because even though they managed to score with their first chance, we showed great character as a team and managed to beat a very good team like Everton," said Mazzarri.
"We have worked a lot on set-pieces. This is another thing we're happy about.
"It was extremely important, because we came from two games we didn't deserve to lose. This gives us a lot of conviction and helps us be concentrated for the upcoming games."
On the performance of Okaka, he added: "He had to wait mainly because of his injury, but knows he showed how important he is to the team. If we would have had him before maybe we would have some more points at the moment, but we are very happy for him."