Suarez opened the scoring only for Jan Vertonghen to score twice – with both goals created by Gareth Bale – within a few minutes either side of half-time to turn the match.
However, a bizarre back-pass from Kyle Walker allowed Stewart Downing to equalise and when Suarez was fouled by Benoit Assou-Ekotto, Steven Gerrard stroked home an 82nd-minute penalty winner.
The Premier League's leading scorer took his tally to 22 in the top flight – 29 for the season – but asked whether that just made him more attractive to Europe's big clubs in the summer, Rodgers agreed, saying: "Football is worldwide now so everyone has seen how good he is.
"But he is very happy among his mates here and you have seen today it doesn't matter where he plays – he was No.10 and then he went in from the side and still created havoc. He is at the top of his game."
Liverpool recorded three successive league victories for the first time since May 2011, but it was as much about character as performance for the Reds boss.
"It was very important win for us," said the Northern Irishman, whose team also beat a top-four side for the first time this season.
"We are on a great run at the moment and we knew this was going to be a severe test of our qualities and character against a team in great form."