The Hornets put in a disappointing first-half display at The Riverside but remained in the contest thanks to a staunch rearguard action.
And the visitors took full advantage thanks to Matej Vydra's brace, the first coming just before half-time before a second seven minutes from the end.
Scott McDonald grabbed a consolation for Boro with virtually the last touch of the game, but it was Watford who headed back to Hertfordshire with all three points to strengthen their grip on sixth place in the Championship table.
"We showed our quality after half-time, but before that we had to rely on our defenders to make sure we still had a chance to win the game," Abdi told the club's official website.
"The first half was very tough. Our manager told us about how Middlesbrough like to play at home and that he believed the match to be a six points type game because of the teams' placings in the division.
"Our defenders won it for us. We couldn't have scored the goals to make it matter if they hadn't played as well as they did.
"In the second half we did very much better and I thought we deserved the win when you take the whole game into consideration."