The Reds had looked to be heading out of the competition with four minutes of their fourth-round replay at the Macron Stadium left after Eidur Gudjohnsen had put second-tier side Bolton ahead from the penalty spot.
But Raheem Sterling equalised before Coutinho, who signed a new contract which will keep him at Anfield until 2020, produced a sublime finish from 25-yards which gave Bolton goalkeeper Andrew Lonergan no chance.
Uruguay striker Suarez inspired Liverpool's Premier League title challenge last season, with 31 goals in 33 matches, to wide acclaim before he left for Barcelona in July last year, and the Liverpool boss claims 22-year-old Coutinho could follow in Suarez's footsteps.
"You would just pay money to watch that kid," Rodgers said.
"He is a great role model for lots of technical players in this country, young British players that for maybe many years would have been discarded because they were technically good but not as big a stature.
"Philippe will become world class in the next couple of years. Luis Suarez was at a level then he played in this team and grew and grew with the model and went into the world-class bracket and I can see him falling in the same sort of way.
"He might not be as prolific as Suarez was but he is certainly on the way to that Liverpool. He is a wonderful, wonderful player. Signing a new deal shows he is really committed to Liverpool and his development."
Liverpool's victory, in Steven Gerrard's 700th career appearance, ensured the Reds kept alive the dream of the Reds captain lifting the FA Cup in his final act with the club before he leaves for Major League Soccer outfit LA Galaxy when his contract expires.
And Rodgers claimed that even though his side trailed to Gudjohnsen's penalty, conceded by Martin Skrtel for a foul on Zach Clough, he always felt they would progress to face Crystal Palace at Selhurst Park in the next round.
"At 1-0 I expected us to win it. The confidence at the moment is very high," added Rodgers.
"Our composure to falling behind was very good. We scored two, could have had many more.
"We have got a great habit at minute which is winning games. You can't question the character.
"It is a great thing to face Palace again. We didn't play anywhere near the level in our defeat their earlier in the campaign.
"That was the lowest moment of the season in terms of performance levels. We go back there in different frame of mind and big confidence."
Bolton manager Neil Lennon risked the wrath of the FA after he was vociferous in his criticism of referee Roger East's performance.
Bolton midfielder Neil Danns was sent off by East for a second bookable offence following a foul on Liverpool midfielder Joe Allen.
"I thought he (East) was rubbish," said Lennon, who has revived the club's fortunes since taking over at the helm in October.
"The first booking Neil gets isn't a booking. He doesn't make contact.
"I thought he spoiled a very good game. I thought things in the game went Liverpool's way.
"The sending off changed the game. The pressure came with being a man down against a great team.
"I'm not convinced that our penalty was a penalty either. I'm not happy with the referee tonight at all."
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