Everton moved to within one point of Champions League qualification with goals from David Weir and Tim Cahill.
Weir headed home Mikel Arteta's free-kick two minutes before the interval after Newcastle missed a host of first-half opportunities.
Barry Knight, who sent off three Newcastle players against Aston Villa, showed Shola Ameobi the red card after 57 minutes for a clash with Cahill.
And Cahill then raced clear to lash home two minutes later to seal victory.
Everton's win leaves Liverpool needing to win at Arsenal on Sunday to stop their Merseyside neighbours clinching fourth place.
And even if Liverpool are victorious at Highbury, Everton will then need only only one point from their final two games.
Newcastle kept Alan Shearer on the bench, while Cahill recovered from a back injury for Everton.
And it was only the lack of a cutting edge that prevented Newcastle from going ahead in a first-half they dominated.
Everton posed the first moment of danger when Arteta's free-kick was brilliantly turned over by Shay Given after seven minutes.
Newcastle then took over, with only poor control letting down Darren Ambrose as he broke clear on Nigel Martyn.
Patrick Kluivert then headed wide from point-blank range from James Milner's free-kick.
And as Everton hung on, Martyn saved well from Milner before Moyes' side struck two minutes before the interval.
Newcastle were furious when referee Knight punished Jean Alain Boumsong for a foul on Marcus Bent.
And their frustration increased when Arteta's free-kick found Weir unmarked at the far post to head home.
Newcastle were reduced to 10 men after 57 minutes when Ameobi lashed out at Cahill when they tangled on the touchline.
And two minutes later Cahill piled on the agony by racing into the area and lashing high past Given.
Everton then held on comfortably to possession as Newcastle failed to pose a serious threat.
Everton boss David Moyes:
"It was terrific. We've played better and at times we were a bit nervy but we got the second goal and that saw us through.
"All in all we're pleased with the three points.
"We've been fourth all year. We know we're in with a great shout. It would be nice if we could celebrate it but we can't because we're not there yet."
Newcastle boss Graeme Souness:
"We feel hard done by again because of Mr Knight.
"I won't defend Shola Ameobi for what he did. When he was sent off the game was over but Boumsong says it wasn't a foul when they scored their first.
"He's given a free-kick to Everton that wasn't and it changed the game."
Everton: Martyn, Hibbert, Yobo, Weir, Watson, Carsley, Bent (McFadden 74), Cahill (Stubbs 89), Arteta, Kilbane, Ferguson (Beattie 63).
Subs Not Used: Wright, Pistone.
Booked: Carsley, Cahill, Arteta.
Goals: Weir 43, Cahill 59.
Newcastle: Given, Carr (Ramage 45), Boumsong, Bramble, Babayaro, Ambrose, N'Zogbia, Jenas, Milner (Shearer 82), Ameobi, Kluivert.
Subs Not Used: Andrew O'Brien, Harper, Robert.
Sent Off: Ameobi (56).
Ref: B Knight (Kent).