Dragic fell a rebound shy of what would have been his second career triple-double with 31 points, 12 assists and nine boards. He was a point short of his career scoring high.
Dragic's only career triple-double came with Houston two years ago and he knew he must have been close to one as this game progressed.
"Somebody told me I had 8 minutes to get one rebound," he said. "I said, 'Don't tell me that.' Then I was thinking too much. It's really tough with those big guys to get one. I was just hoping maybe to get one long rebound that's easy to grab. But you know in the end, they executed well. They didn't miss a lot of shots. Deron Williams, in the low post, he went to the free throw line, a layup, a 3-pointer. So it was a lot of good decisions from them."
Brook Lopez added 20 points and Kris Humphries a season-best 17 for Brooklyn. Watson scored 12 of his 14 points in the fourth quarter for the Nets, playing without Joe Johnson, who bruised his thigh in Brooklyn's loss to the Clippers in Los Angeles on Saturday night.
"On the road and back to back you have to find a way to win. Not taking anything away how well they played -- Joe Johnson didn't play," Nets coach P.J. Carlesimo said. "It is still a very good win. It wasn't pretty. It wasn't like we are going to say we did 800 things well but we did enough things to win. Maybe we didn't play as well as we did last night and we got a loss last night. Tonight we found a way to get the win."
The Suns had a chance at the end when Dragic purposely missed the second of two free with 3 seconds left and Phoenix's Hamed Haddadi controlled the rebound. But the big Iranian's shot was off the mark.
Brooklyn improved to 3-1 halfway through eight consecutive road games.
"A win is a win. We will take it," Williams said. "It is a good start (on the road). It wasn't the way we wanted to win it but it is a win none the less and puts us 3-1 which is pretty good on this road trip."
Wesley Johnson added 21 points -- including 5 of 9 3-pointers -- for the Suns, who were coming off consecutive blowout losses at home to Washington and Minnesota.
Down by 16 early in the third quarter, the Suns mounted a comeback behind Dragic's floor play and the shooting of Johnson, who finished the period with four consecutive 3s to put Phoenix up 76-75 entering the fourth.
Johnson's first 3 in the spree cut it to 69-67, his second made it 71-70, his third 74-73 and his fourth, with 8.1 seconds left, gave Phoenix its one-point lead after three.
After Dragic's two free throws tied it at 87-87 with 5:53 to play, Williams' 3-pointer put Brooklyn up for good 90-87 with 5:08 to go.
Williams' driving layup made it 99-95 with 2:12 to play, then Johnson's 22-footer sliced it to 99-97 with 1:45 remaining.
Phoenix had several chances after that, but missed a trio of shots in a row after two offensive rebounds, the last a 16-footer by Dragic with 27 seconds left.
Gerald Wallace, with cotton stuffed into his right nostril, missed the first of two free throws but made the second and the Nets led 100-97 with 14 seconds to play.
Dragic drove for two to make it 100-99 with 6.8 seconds to go. Watson made two free throws to put Brooklyn ahead 102-99 6.3 from the finish. Dragic was fouled with 3 seconds left, made the first free throw, then lined the second off the rim. The 7-foot-2 Haddadi got the rebound but his shot was too hard off the backboard and the Nets escaped.
"We are of course disappointed because we lost," Suns interim coach Lindsey Hunter said, "but if we can have that type of every night we will take it."
Brooklyn finished the first half on a 15-7 run, capped by Humphries' three-point play with 23.1 seconds left, to lead 55-43.
Williams scored the first four points of the third quarter to stretch the lead to 59-43, the Nets' largest of the night.
Dragic reached double digits in assists for a career-best fifth game in a row. His 11 double-digit assist efforts since the All-Star break are second only to Milwaukee's 12 and Jennings has played one more game. ... The Nets' eight consecutive road games match a career long. They will be on the road for seven straight games, then return home before traveling to the eighth game, April 3 at Cleveland. ... Brooklyn's next home game is April 4 against Chicago. ... Phoenix was without C Jermaine O'Neal because of a calf strain. ... Hall of famer Julius Erving had a courtside seat.