As well as Cronk, the Storm were without playmakers Cody Walker (hamstring), Joel Romelo (ankle) and Ben Hampton (suspension).
And it could get worse for the Storm, with winger Young Tonumaipea placed on report for a suspected crusher tackle on Tate in the 34th minute.
Tate was at the centre of a match review controversy during the week after being cartwheeled in Origin by Blues player Josh Reynolds who escaped suspension.
Tate was left clutching at his neck after Tonumaipea twisted it in a tackle.
The Cowboys surged to a 16-0 halftime lead with Kane Linnett, Antonio Winterstein and Thurston all crossing as the weight of 64 per cent possession took its toll.
Thurston iced the game in the 65th minute, kicking the ball through the line before toeing it under the sticks for fullback Michael Morgan to pounce.
Scott (164m) and Tamou (121m) were tremendous backing up from Origin, racking up big metres in a Cowboys pack that also featured dominant performances from Scott Bolton (154m), Jason Taumalolo (159m) and Ashton Sims (123m).
Young fullback Cameron Munster was called on by the Storm to fill the sizeable boots vacated by Slater and was pressured early, coughing up the ball three times.
2013 NYC Lock of the Year John Asiata made his NRL debut for the Cowboys after following former Roosters assistant Paul Green to North Queensland in the off-season.
-AAP