Defeat to Besiktas in the UEFA Europa League on Thursday meant Rodgers' squad were not back in England until early on Friday, limiting their preparation time for Sunday's Premier League clash.
But tiredness did not impact the hosts as stunning strikes from Jordan Henderson and Philippe Coutinho lit up Anfield on Sunday.
"I have to take my hat off to my players," Rodgers told Sky Sports. "To get back at half four in the early hours of Friday morning - by the time they got to bed it was six, and they had to sleep all of Friday.
"We then did some light work yesterday, but to come out and play like they did, they were relentless.
"They restricted a top team to just one shot on target and deserve all the praise they will get.
"The character of this team was questioned at the beginning of this season, but we were just trying to find a rhythm and balance to the team, with little time and lots of new players. It hasn't just happened; they have worked hard, and believed in how they were working."
Victory keeps Liverpool firmly in the hunt for a top-four finish, with their unbeaten league run now standing at 11 games, but Rodgers is not getting carried away ahead of Wednesday's visit of Burnley.
"We just need to concentrate on Burnley now," he added. "Since November our points and performance levels have been very good.
"I believe if we get into the top four it will be a better achievement than doing it last season, but we have got to keep our focus and concentration."
