Steven Gerrard converted from the spot in front of the Kop to banish memories of his penalty miss against West Brom on Monday night.
Coutinho weighed in with a second after the interval before Jose Enrique, Luis Suarez and then Daniel Sturridge, also from 12 yards, completed the rout.
"It was an important three points," said Rodgers. "We've been down a bit of a rocky road this week in terms of the games on Monday and Thursday.
"But the Premier League is very important to us and we knew we had to go out with commitment today against a Swansea team that has been brilliant this season.
"It was an outstanding performance and we really showed our character and mental toughness today to get a performance like that. Going into the game it was about keeping a clean-sheet because in the last four games, we haven't been at our best defensively. So I was very pleased.
"We looked to dominate and create chances and we did. Like in any game of football, it's important that you start the second half well and in that little period, we scored three goals and that really gave us extra confidence."