Paul Hill's second-half score put the black, green and gold ahead as they sought to bounce back from the previous weekend's 31-7 loss to Gloucester.
But the Falcons responded strongly, scoring twice to send the majority of the 1,750 supporters in attendance home happy.
Saints were unable to even take a losing bonus point from the match, meaning they remain 11 points behind fourth-placed Harlequins with just three games to go.
And when asked how he saw Saints' top-four situation now, Boyd said: "We're dead, mate - we're gone.
"Somebody would have to lose their last three games for it to happen and if you play like we have in the past two weeks, you don't deserve to be in the top four anyway.
"We're massively disappointed around the fact we set ourselves up nicely and have had a pretty tough run.
"All we're worried about now is that we've got three pretty tough games to finish off the season: Wasps and Exeter at home and then Bath away.
"We need to finish the season with some pride."
On the performance at Newcastle, Boyd said: "It was 3-3 at half-time and we just weren't able to get our game going because we just kept doing the wrong things, decision wise and skill execution wise.
"It took a lot of juice out of our legs in that first 40 but when we went 10-3 up I thought 'if we can manage the game we can bring it home and put them under pressure', but we didn't.
"We overplayed through the middle of the park and didn't do the things we talked about so we paid the price."