Liverpool finished last season with a shambolic 6-1 drubbing at the Britannia Stadium but gained a measure or revenge at the weekend thanks to Coutinho’s long-range strike.
Referring to Coutinho’s intervention, the former Manchester City man said: “I obviously saw him do it a few times last year – so to do it when he was on my team was nicer!
“That’s what the game needed. It was a tough game, a battle; I thought we defended really well. It was going to take a moment of brilliance to win the game and we managed to get that.
“Not just being the first game of the season, but obviously with what happened in the last game of last season, the boys showed great character.
“Any time you can walk away from here with three points, it’s a great result.” Milner was one of four summer signings to start Sunday’s game.
Christian Benteke, Nathaniel Clyne and Joe Gomez were also named in Rodgers’ line-up, while Roberto Firmino was introduced during the second half.
Milner was hugely impressed with the quality of Liverpool’s defending.
He added: “We have got a lot of quality going forward, but a lot of credit goes to the whole team and the back four.
“Clyney looks like he’s been here for 10 years; Joe coming in for his first game in the Premier League playing against [Jonathan] Walters, who is an experienced pro, very clever and a tough player to play against – I thought he was outstanding and did a great job.
“The two centre-halves were brave and kept a really good line. That makes all the difference for us as a midfield and a team, keeping us high.”
Firmino, meanwhile, scored a hat-trick in a behind-closed-doors friendly against Welsh club TNS at the club’s training ground yesterday.
The Brazil international is some weeks behind most of his team-mates in terms of fitness having joined pre-season late because of the Copa America.
Firmino made a 12-minute cameo at Stoke – but manager Brendan Rodgers felt the 23-year-old and other players short on match practice needed extra work.
Those included Liverpool’s other goalscorers in a 5-1 win – Danny Ings and Lazar Markovic.
Emre Can, who also returned to Melwood later after playing in the European U21 Championships but played for 27 minutes at the Britannia Stadium, also featured along with Adam Bogdan, Lucas Leiva, Alberto Moreno, Divock Origi, Jordan Rossiter, Mamadou Sakho and Kolo Toure.