Cincinnati backup Quinn Sharp kicked field goals of 34 and 51 yards after that for the deciding margin.
Both teams played their starters the entire first half.
Palmer was facing the team he quarterbacked for the first eight years of his NFL career before an acrimonious split sent him to Oakland. Dalton replaced Palmer as a rookie in 2011.
Cornerback Leon Hall said "it means something to me, for emotional reasons I guess" to hold Palmer, his former teammate, without a touchdown.
"But really," Hall said, "I'd feel almost the same way regardless who we were playing. This was what they call a rehearsal game. ... This builds some momentum for (the opener against) Baltimore."
Cincinnati inside linebacker Vontaze Burfict left the game in the first quarter with a hamstring injury.
The matchup featured two of the top defenses from last season.
On the Cardinals' second possession, it was third-and-7 from the Arizona 40 when Palmer fired a pass intended for Fitzgerald. But the errant throw went right to Newman, who evaded a couple of early would-be tacklers and outran everyone else to the end zone.
Statistically, at least, the Bengals' starters had the advantage, outgaining Arizona 198-141 in the first half.
"On defense, we had some negative plays," Cincinnati coach Marvin Lewis said, "but overall the effort, and of course the point total, was what you are looking for."
Cincinnati led 13-3 at the half.
NOTES: Bengals rookie guard Trey Hopkins was carted off the field in the fourth quarter with a right shin injury. Moments later wide receiver James Wright left via cart with a concussion. Guard T.J. Johnson became the third Bengal carted off the field in the fourth quarter. ... Arizona was without starting inside linebacker Kevin Minter (pectoral muscle) and guard Jonathan Cooper (toe).
Copyright 2014 by The Associated Press