The Uruguay international returned to training with the first-team on Friday having been exiled for more than a week after criticising the Reds boss, who said his attitude and performances had not been up to standard.
Suarez claimed the club had reneged on a deal agreed last summer to allow him to leave if a Champions League club came in for him, with Arsenal having had two bids for him rejected.
"I am satisfied with how everything has been resolved and we as a club are looking to the future," said Rodgers, who refused to confirm whether the player had apologised as he said he would have to in order to be reintegrated.
"There is nothing to be said there. I won't be saying anything on Luis Suarez and that is only purely out of respect for the (other) players."
Suarez is still suspended for a further six matches for biting Chelsea's Branislav Ivanovic in April, but he was at Anfield to see his side win 1-0 against Stoke.
His pre-match walk around the pitch's perimeter brought muted applause and not the rapturous welcome he had received when he played in Steven Gerrard's testimonial a fortnight ago at Anfield.
"He is a part of our team and club and the supporters are a class act here," said Rodgers.
Rodgers had earlier described Suarez as "back...smiling and happy" in his pre-match interview with BT Sport.
Suarez went public with his grievances last week after a second bid of £40,000,001 by Arsenal, which they thought should have triggered his departure, was rejected by the Merseysiders.
Liverpool have always maintained there was no release clause as such, only a requirement to notify the player and begin negotiations should they wish.
Suarez seemed much happier before Saturday's game however, carrying daughter Delfina around the edge of the pitch to a ripple of applause from the smattering of fans assembled inside the ground an hour before kick-off.
He even stopped to shake hands with former referee Mark Halsey at pitchside before disappearing down the tunnel, emerging less than 10 minutes later to make the walk back to the executive boxes in the Centenary Stand on the opposite side of the ground.
A small group of Stoke bans booed as he walked past them but there was more applause and the odd cheer.
Rodgers paid tribute to Simon Mignolet after the goalkeeper saved a late penalty to secure the win.
Daniel Sturridge scored the only goal shortly before half-time but after missing a number of chances they were indebted to their £9million summer signing from Sunderland for saving Jonathan Walters' 87th-minute spot-kick.
Mignolet's signing was questioned by many, mainly because it hastened the exit of established first choice Jose Reina who left on loan to Napoli soon after, but he has certainly begun well in his bid to win over the doubters.
"It was a terrific performance all-round but of course the goalkeeper gets the plaudits when he saves a penalty at the end," said Rodgers.
"He made three terrific saves in the game. He made one in the first half from Walters but obviously when you don't take your chances - and we created numerous chances - and you get a penalty against you at the end you have that sinking feeling.
"But he had done his homework, he made a good first save and his second action to get up and make the double save (from substitute Kenwyne Jones) was impressive. From that (save) you also saw the spirit in the team and what we are trying to cultivate here."
Stoke's assistant manager Mark Bowen stood in at the post-match press conference as boss Mark Hughes had dashed off to scout a player somewhere in Europe.
"You're disappointed because you think you're going to come away with a point," he said of Walters' missed spot-kick. "He's missed a couple in the past but he struck it well enough. I just thought the keeper made a good save.
"Jon Walters gives everything for this club, so nobody in the dressing room will do anything other than pick him up and get him ready for the next game."
Bowen had praise for his own goalkeeper Asmir Begovic, who did more than enough to give the Potters a chance of snatching a point with a string of saves.
"I thought it was going to be a battle and the keeper (Asmir Begovic) kept us in it at times," he added.
"It was a case of staying in the game as long as you can because we are asking them to play a different kind of game and you hope they don't lose confidence, which I didn't think they did."
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- Luis Suarez