Coutinho, an £8.5million January signing from Inter Milan was making only his third appearance for the club, but provided assists for the first two goals, crossing for Downing and then sliding the pass through for the first of Luis Suarez's hat-trick in an encouraging 70-minute display.
Although Downing believes the 20-year-old will take time to adapt to the English game, he is heartened by what he's seen from the Brazilian so far.
"He's had two good games (his other match was a 12-minute debut as a substitute)," he said. "He will find it hard physically because this is different from the two leagues he's played in.
"I think he's settled in well and he's had two really good performances.
"If he can keep playing like that - and with Luis banging in the goals - then we've got a really good chance."
Downing also believes goalkeeper Jose Reina is returning to top form again after the Spain international played a crucial role in the victory, pulling off brilliant saves from Emmerson Boyce and Franco Di Santo at 2-0 and 3-0 in the first half.
The Spaniard has been criticised for errors this season but his team-mate feels the goalkeeper is back to his best.
"It's hard for keepers. As players, we make many mistakes in games, but they get overshadowed by other things," Downing said.
"A keeper makes a mistake and it's in the back of the net. It is a tough position to play.
"But you see his save at 2-0 and if we had gone in 2-1 at half-time that would have made it more difficult.
"People don't tend to see that, but we as players could see it was an important save.
"It gave us the platform to go on and get the third goal and it was easy from then on.
"He's a top-class goalkeeper. Everyone has dips in form and things like that.
"It's about the way you bounce back and he certainly has."