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 around.
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 Suarez 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 said: "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 [Sunday] it doesn't matter where he plays - he was in the number 10 role and then he went in from the side he still created havoc.
"He knows the trust I put in him and that gives him the platform to go and perform."
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.
"For us to get three points and show the character was fantastic and I am delighted for the players because they worked ever so hard.
"You can see the mentality in the group. I thought we had to show different characteristics today [Sunday] than what we have had to do in recent weeks.
"We've been comfortable in a lot of the games and been able to control them but against a top side we came through.
I had to make some changes in the game to get the control back in our favour and when we did that I thought we were good value for the win."