The forward's goal kept the Merseysiders within touching distance of the Premier League top four and diminishes City's own hopes of retaining their crown with them five points behind Chelsea having played a game more.
But it was Coutinho's strike from the left corner of the penalty area 15 minutes from time that continued his personal rise.
The 22-year-old, who arrived from Inter Milan in 2013 for £8.5 million signed a new five-year contract last month and has added goals to his game following Rodgers' request.
The Liverpool manager also told how this victory was particularly pleasing given the late night trip home from Turkey where his side were knocked out of the Europa League to Besiktas on Thursday.
"It was a brilliant result and an equally brilliant performance," said Rodgers
"We thoroughly deserved it, the players were relentless. It was very pleasing.
"Getting back at 4:30am in the morning on Thursday and not getting home to 6am and then to prepare for a game against the champions is not easy. You have to take your hat off to them for the way they dealt with that.
"It's a big one but there are still a lot of games to go. It was a significant win because it keeps us in the chase.
"Coutinho is operating on a very high level. His performances in particular since he signed his contract have been a joy to watch.
"He came in at 19, is still developing. He's a sensational footballer. To beat Joe Hart from that angle with that power was phenomenal.
"He is a player that has always assisted and made the final pass, he's a very selfless player, he's a very humble guy. There are improvements he can make, he is playing at a football club who adores him and loves it here.
"He has a humility, wants to work and is an absolute joy to work with. He was £8.5 million a few years ago, it's frightening to know what he's worth now.
"To keep them to one shot on target in a game they wanted to win was incredible.
"We can only focus on Burnley, we just have to concentrate on the next game and finish as high as we possibly can.
"If we can get into the top four it is better achievement than last season. We just need to keep collating the points."
Source: AFP