Cilic wasted four set points when Pouille battled back from 0-40 down at 2-5 in the second set but with his own serve rock-solid he calmly moved into a two-set lead.
The mainly French 22,000 crowd kept roaring their support for Pouille, but it was the Croatia contingent, decked out in their distinctive red and white, who would soon be celebrating.
Cilic broke for a 3-2 lead in the third set and then closed in for the kill when Pouille served at 3-5. The Frenchman saved two match points from 0-40 but Cilic would not be denied and finished it off with a lob.
The 30-year-old was immediately swamped by his team mates as Croatia emulated their triumph of 2005.
Cilic said: "For us, it's a dream come true for this nation. We are so passionate. You can see the fans, and I feel that in Croatia it's going to be incredible too."
Croatia's win will enter the history books as the last final played before the 118-year-old team competition gets a revamp. Next November a new format will see 18 teams assemble in Madrid for a World Cup-style event to decide the champions.
Davis Cup final - France 1-3 Croatia
Borna Coric (CRO) bt Jeremy Chardy (FRA) 6-2 7-5 6-4