Jackson walked eight, all but one in the first three innings, and threw a season-high 149 pitches in the second no-hitter in Diamondbacks' history. This is the third time the Rays have been no-hit since last July, including Dallas Braden's perfect game at Oakland on May 9.
Jason Bartlett grounded to shortstop for the final out and Jackson's teammates mobbed him on the field.
Jackson pitched for the Rays from 2006-08. His only other shutout in 126 career starts came for Tampa Bay in 2007.
"To be able to come back and give the crowd I used to play for something to see," Jackson said. "Today is just a special day. Something I'll never forget."
Reynolds made another big play on a sharp line drive by Bartlett in the seventh. In the eighth, Miguel Montero threw out pinch-runner Carl Crawford trying to steal second after Carlos Pena reached on an error.
Jackson was so wild and the pitch count unusually high that the Diamondbacks had relievers warming up from the sixth inning on.