Reading have not won at the Madejski Stadium since January and were undone by Martyn Waghorn's 45th-minute header. The forward, on loan from Sunderland, bravely converted a Matt Oakley corner to hand the visitors a hard-fought victory.
"I hope I am given time," said Rodgers, who left Watford to take over the reins in Berkshire. "I've been given a three-year contract but results aren't great, I understand that.
"But that was a good performance.
"I won't feel sorry for myself. I have 100 per cent belief in my ability and I remain confident that we can turn it round.
"We played some terrific football but I am very proud of the players.
"There was a lot of pressure on them but they were very good and didn't go missing.
"I try and take the positives from every performance and the public can see that there is lots of great things about our game."
"There is criticism and it's not easy. I live in Reading so there is no escape.
"I started badly at Watford but at least I could drive away at the end of each day. I was probably getting the same stick but I couldn't hear it.
"I try not to read the papers and listen to things but when it's your town you can't get away from it.
"It's only a matter of time before we change the atmosphere but we need results."
Nigel Pearson paid tribute to keeper Chris Weale and his defence as his side rode their luck to clinch three hard-earned points.
"I'll be the first to admit that our goal at times had a charmed life," he conceded.
"We were playing a side who were very low in confidence and they were certainly unfortunate.
"Clean sheets are pleasing but we have to play better than that.
"When you don't play well you have to dig in and I thought that even though we weren't at our best we were prepared to stick in there.
"You can't preen your feathers with a win like this but it could prove to be a fantastic result for us."
Reading were guilty of wasting a string of chances with Simon Church the main culprit, although Jobi McAnuff and Grzegorz Rasiak also had opportunities for the home side.
McAnuff managed to beat Weale but saw his shot hit a post and, after dominating much of the half, the Royals instead went behind when Waghorn beat Gulfi Sigurdsson to nod past Federici with an accurate finish.
Federici denied Matty Fryatt but Reading created further openings and Rasiak could have done better from a pass by Jimmy Kebe in the closing stages.
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