Allan Saint-Maximin gave the Magpies the lead early in the second half, but a late equaliser from Joao Pedro meant the points were shared.
A win would've dragged them out of the drop zone, but Saturday's draw means they stay 19th.
With just one win so far this campaign, Newcastle are running out of time if they are to stay up this season, but Eddie Howe is refusing to throw in the towel.
"We saw today as a huge opportunity and that's why it hurts so much. We were really pleased with what we saw up until half-time, so that gave us even more optimism that we could go on and win this match.
"One-nil for me is not the time to go into our shells. For me, that's the time to put the game to bed, that's the best way to handle that situation.
"But we have to understand the players' mentality at this moment in time, maybe not thinking the same way, so that's something we can change.
"There are a lot of twists and turns to come. We still believe in ourselves and the players, but obviously we know that every game gets more and more important."
Newcastle looked the better side for much of the first half, but had to wait until after the interval to claim their much-deserved lead.
However, after Saint-Maximin found the net, Newcastle dropped further and further back as the clock ticked on.
Eventually their negative approach cost them a vital two points, with Pedro's late goal denying the hosts the chance to leapfrog their opponents.
It's an issue Newcastle have struggled with this season, having now dropped a huge 21 points from winning positions.
But despite the frustration, Howe insisted his side have to stay positive if they are to turn their results around.
"I don't think it's the time for shouting and screaming and that kind of reaction.
"I don't think the players would welcome that and I don't think that would do any good apart from just airing my frustrations on them and I don't think that's healthy with the current league position.
"I think it's a case of trying to help the players understand why we didn't win and then showing them and doing the work on the training pitch. I think that's the only process that's going to work."
Newcastle now face another huge match with Leeds United next Saturday, with the Yorkshire-based side also struggling in the bottom half of the table.