Palace's 12-match winless Premier League run cost Patrick Vieira (46) his job last week and the club coaxed Hodgson out of retirement in a bid to keep them away from the relegation zone.
It paid instant dividends as Jean-Philippe Mateta's (25) 95th-minute winner secured an invaluable three points with a 2-1 home victory against fellow strugglers Leicester CityLeicester City.
"It's a wonderful feeling," Hodgson, who was in charge of Palace between 2017-2021, told reporters.
"It's such a relief, really, because I thought over the course of the 90 minutes we deserved to win the game - especially with the first-half performance.
"As the game wears on, we conceded a very good goal to them and you can see the game petering out at 1-1. I honestly believe today, with everything we did on the field and the stats back it up, we deserved to win it."
The victory put Palace up to 12th although they are still only five points above Leicester who dropped into the bottom three.
For a side that has struggled to score goals, Palace's attacking statistics were impressive with 31 attempts at goals compared to just three for Leicester.
"It is quite incredible to have that amount of shots, that amount of shots on target and that amount of corners," Hodgson said. "It really did show how intent we were to play the game in their half and when we got to the final third to profit and capitalise on that."
While there was joy for Palace, Leicester's situation is looking bleak. They have picked up only one point from the last 18 on offer and the pressure is mounting on manager Brendan Rodgers (50).
He insisted, however, that now is not the time to panic.
"Everything's in our control. We have a lot of points to play for," Rodgers, whose side take on Aston Villa in midweek, told reporters. "We've been in the position before and dealt with that adverse moment and got ourselves out of it.
"The confidence is not quite where we want it to be, but the players are working hard."
