Adam Swift crossed for the first try in the 11th minute but Saints, who conceded only twice in their opening two Super League games, had to absorb their fair share of pressure in remainder of the first half.
Mike McMeekan looked to have put Tansey through with his first touch of the ball but the pass was forward.
The 24 year-old, who spent most of last season on the sideline, took a short ball from Travis Burns to slip through a gap before looping the pass to Tommy Makinson for 10-0 at the break.
But the big Australian winger more than made up for his error after touching down for a quickfire double to put the Tigers back within six points.
Cumbrian forward Kyle Amor was sent to the sin bin for a dangerous tackle but Saints rallied and put the game beyond doubt, scoring through Mark Percival who in turn converted from the touchline for 20-8.