KR, who went into the game six points behind leaders Leeds Rhinos, started well with Ken Sio scoring an early try for the visitors.
Tansey then turned provider when his pass which took a ricochet and fell into the hands of Luke Gale, who crossed over the whitewash to open up a 12-4 lead.
The hosts extended that advantage further before the interval when Nathan Massey touched down.
Gale added a drop goal early in the second half before Cas made sure of victory when Frankie Mariano scored a try 15 minutes from time.
The Tigers now leapfrog KR in the table, while the Humberside outfit drop to sixth.