It lifted the Dragons above Glasgow Warriors in the table, while the luckless Italians remain rooted to the bottom without a win in eight attempts.
The home side's hero was their fit-again fly-half, who kicked well in front of watching Wales coach Warren Gatland.
Tovey's five penalties and two conversions came in his first game for more than a month, after suffering a torn hamstring.
The Dragons relied heavily on the boot of Tovey, the fly-half booting three penalties in the opening 20 minutes - the third after Italian prop Redolfini was yellow-carded for preventing release.
That was quickly followed by a cracking try started and finished by Wales candidate Aled Brew, with Tovey acting as the middle man. Brew chipped ahead, Tovey neatly kept the ball in play and dashed infield, where Brew took his pass and swept over for a try which Tovey converted.
His fourth penalty after half an hour put the Dragons well ahead, but Aironi replied on the stroke of half-time with a try in the corner by winger Giovanbattista Venditti after a series of drives by their big pack.
The Dragons led 19-5 at the interval and Tovey was soon on the mark again, landing another penalty - although the Dragons were down to 14 men when hooker Steve Jones was yellow-carded for coming in from the side.
The Italians had two more players yellow-carded for persistent infringement, both of them replacements - fly-half Riccardo Bocchino and hooker Roberto Santamaria, his departure being a second yellow which turned to red and saw them reduced to 13 men.
The Dragons wrapped it up with six minutes left when Brew's fellow wing Adam Hughes romped clear for a try which Tovey converted.
Deep into injury time Hughes grabbed his second try after chasing a fly-kick, replacement Matthew Jones converting - although the Dragons did miss out on a bonus point.
|Newport Gwent Dragons Score Card|
|Aironi Rugby Score Card|