The Argentina international set up Luis Suarez for an equaliser after Girona had taken a third-minute lead, and netted twice himself, the second an inventive free-kick steered under the opposition's defensive wall at Camp Nou.
Coutinho, a January arrival from Liverpool, also supplied an assist for Suarez before getting on the scoresheet, and the Brazilian is already revelling in a productive relationship with his superstar team-mates.
"Leo is amazing," he told Movistar.
"He fooled everyone with his goal. It's incredible how good he is. Each game we understand each other better, with him, with Suarez, he's very important for us.
"Bit by bit I'm learning and I have a lot of work to do."
Holding midfielder Busquets echoed the sentiments of Coutinho, saying of Messi's contribution: "He always makes the difference and that's what makes him the best player in the world and in the history of football."
Ousmane Dembele could not find the back of the net himself but did tee up Suarez for the Uruguayan to complete his hat-trick, and Coutinho expects the young France international to hit his stride soon.
"Barcelona play unique football," he said.
"Dembele is a great player, I'm sure that with time and work we will learn. We want to do the best we can."