Palace were in front thanks to James McArthur's first-minute goal when Niasse went down theatrically under pressure from Palace defender Scott Dann, with referee Anthony Taylor pointing to the spot.
Leighton Baines converted from 12 yards, then Niasse came up with a second Everton equaliser after Wilfried Zaha had put Palace back in front, leaving Hodgson despondent as rock-bottom Palace's winless run was extended to five games in all competitions.
"We feel hard done by, I suppose," Hodgson said. "I don't think we could have played a lot better.
"The goals we conceded were unfortunate - the first was a dive, and the second was our doing as we committed harakiri and it let them back into the game. But we really did boss the game, playing some excellent football.
"We have dropped two more points. Mistakes happen, don't they? Being wise after the event doesn't serve any purpose at all. We have got no one to blame but ourselves.
"I can't ask for more than I got from the players today, but I must admit to being bitterly disappointed. It is not enough just to give players credit for a performance - as a manager I want to see some points on the board."
RH: I didn't think it was a foul [for the penalty]. The only reason it's a penalty is because it was given as a penalty.#CPFC#CRYEVE pic.twitter.com/DFhZ7nDZv9 — Crystal Palace F.C. (@CPFC) November 18, 2017
With Everton still yet to appoint a replacement for Ronald Koeman after the Dutchman was sacked last month, caretaker manager David Unsworth has overseen successive fightbacks to claim four points in two Premier League games.
And a thrilled Unsworth defended the resurgent Niasse, who could face a ban due to new rules allowing retrospective action against players found to have deceived officials.
"That was a well-fought point," said Unsworth. "We had to come back twice and we showed great resilience and desire. I am delighted with a point.
"We conceded two poor goals - our own downfall. To concede so early is always a blow but we came back strongly and this team has shown they have fight and desire.
"It was hard for me to see my team's penalty decision, but the referee gave it. If Niasse exaggerates it then that's the case, but we're just thankful that we got it and Bainesy took it well.
"I worked with Niasse in the under-23s and he gives you everything he's got. He is a pest - a nice pest for us. I am delighted for him, because he is a great guy as well."
Despite making a case to be given the job on a permanent basis, Unsworth remains in the dark about his future amid Everton being strongly linked with Watford's Marco Silva.
"Three thousand of our fans turned out tonight and we gave them something to shout about," added Unsworth. "As for me, I don't know. Tomorrow is a good day off, then we'll see what happens on Monday."