Oakland had until Wednesday to match the Cowboys' three-year offer sheet worth $8.25 million, which includes $4 million in the first year.
Mayowa, who took to Twitter on Tuesday afternoon, had two sacks in 28 games for the Raiders. He also forced a fumble and recorded a safety.
#Blessed lets work— Benson Mayowa (@Benny_b0y10) March 22, 2016
"At the end of the day, you watch him on tape and you love the way he plays," Cowboys executive vice president Stephen Jones said Monday night before the Raiders made their decision. "Obviously we're trying to marry people to what we do on defense and [defensive coordinator Rod Marinelli] and the type of scheme he wants and the players he wants.
"The guy plays hard. He's got the movement that we look for in defensive linemen, and we feel like he's got some upside. Do we think we're getting the next DeMarcus Ware? Probably not. But when you don't have the guy, then you've got to do it by committee and hope you get the right players, playing the right way so that you get pressure."
The Cowboys have made no effort to retain Greg Hardy, who is a free agent. They also have Jeremy Mincey and Jack Crawford as free-agent defensive ends. Randy Gregory will miss the first four games because of a suspension, and last year's leader in sacks, DeMarcus Lawrence, is rehabbing from offseason back surgery.
The signing of Mayowa would not preclude the Cowboys from using the fourth pick of the draft on a pass-rusher.