The Cottagers, without a Premier League away win since September 2006, had the best of the first half with new signing Leon Andreasen shooting just over.
But Bolton should have scored after the break when Lubomir Michalik cushioned the ball wide with the goal gaping.
Fulham are now four points from safety while Bolton slip to 16th in the table.
Hodgson said afterwards: "The point gives me a lot encouragement in terms of the way the team played.
"In the second half our players were tired and we didn't get hold of the ball so much, but we defended very well throughout.
"My group of players are good professionals who want to improve, work hard at their game, and roll up their sleeves and show the necessary fighting spirit to get us through.
"Tonight was a big step forward, we've got 14 games left to go, and we certainly aren't throwing in the towel."
Bolton manager Gary Megson admitted: "The frustration was clear for all to hear from the fans at the end but you could see the Fulham players were delighted to come away with a point.
"We needed to score to settle down a bit but there's going to be a lot of games up and down the country like that at the moment.
"We are down in the bottom six or seven but that might prove to be a decent point at the end of the season, though our quality wasn't there today.
"We have to take a positive from tonight and that's a clean sheet."