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Palace captain Luka Milivojevic opened the scoring from the spot mid-way through the first half after Cyrus Christie handled in the area.
Jeffrey Schlupp then added a late second as he tapped in the rebound following a Sergio Rico's fantastic save to deny deadline-day arrival Michy Batshuayi, who had come on moments earlier.
The Eagles, who rose to 14th, had only won once in their previous six Premier League outings prior to Saturday's victory over Fulham - which was only their third home league win of the season.
"We were not under pressure but these are anxious times," Hodgson told BBC Sport.
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"If we had more points on the board, points that we deserved, it would be less problematic here. Every point is so valuable and if you're in the lead you can't stop playing and worry about the final whistle."
Fulham have now lost 16 of their 25 Premier League games this campaign and they remain 19th - seven points clear of safety..
Schlupp has bagged 4 four goals in 21 matches for Palace in the ongoing season.