The host had only been able to stay in the tie because of victory in the doubles rubber but Cilic made no mistake in the first of two scheduled matches.
Cilic was part of the side that lost 3-2 to Argentina in 2016 showpiece and he was ruthless in ensuring his country did not suffer another heart-breaking defeat.
After edging the first set on a tie-break, Cilic needed just a break apiece in the second and third to secure glory for Croatia, wrapping up a 7-6 (7-3) 6-3 6-3 success with a glorious lob before he was mobbed by his joyous team-mates.
Cilic dictated matters in the tie-break at the end of a first set, in which Pouille twice held after saving break points. The 2014 US Open champion producing a fine off-balance backhand volley to forge ahead.
Pouille played his part with some aesthetically pleasing tennis in the second but he was broken in the sixth game and, despite saving four set points two games later, he succumbed to the power of Cilic, who moved two sets up with a well-struck forehand.
The f's defences were breached again in the fifth game of the third set as, deceived by the bounce of the ball on the baseline, he lofted a backhand long to put Cilic and Croatia on the brink.
A booming forehand gave Cilic three match points as Pouille could only return it into net. Two went begging but Cilic would not be denied a third time as his glorious forehand lob put the disappointment of two years ago to bed.