German forward Woltemade volleyed home Harvey Barnes' cross just before the 15-minute mark to give the hosts the lead at St. James' Park.
And just three minutes later, Osula doubled the advantage with an unerring low finish after some fine build-up play from United.
Four minutes after the hour, Osula rolled home his second of the evening from Joe Willock's low cross to all but seal the points for Eddie Howe's charges on Tyneside.
While Taty Castellanos' sensational long-range effort gave the visitors some hope, United kept them at bay in the closing stages to take the points - and allow Kieran Trippier, who was substituted late on, to enjoy the acclaim of the St. James' Park faithful on his final home appearance for the club.
Trippier started along with Jacob Ramsey and Harvey Barnes as Eddie Howe made three changes from the side that drew 1-1 at Nottingham Forest last weekend.
In need of a win to boost their hopes of avoiding relegation, the Hammers started with urgency but the persistent Osula was a constant thorn in their side early on. The Danish frontman first forced an error from Konstantinos Mavropanos before a neat run a few minutes later brought a shot which was charged down.
And on 14 minutes, Barnes beat Jean-Clair Todibo to a loose ball on the right and delivered a fine cutback for Woltemade, who duly dispatched his 11th goal of his debut campaign on Tyneside.
After a swift, slick move involving Barnes, Bruno Guimarães and Jacob Ramsey, the latter threaded a perfect first-time through ball for Osula and the former Sheffield United frontman finished low through the legs of goalkeeper Mads Hermansen to make it two.
The visitors went close to reducing the deficit just before the half-hour, though, and it took two smart stops from Nick Pope to keep them out. Crysencio Summerville skipped to the byline and his cross found Castellanos - sent on in place of Todibo just after Osula's strike - but Pope did superbly to repel his finish. The rebound eventually fell to El-Hadji Malick Diouf, who forced Pope to parry his powerful effort across the face of goal.
Nuno Espírito Santo's men tried to lift themselves after the restart but struggled to break down a compact United backline. A deep Diouf cross found ex-Magpie Callum Wilson at the back post but Sven Botman did well to recover and clear, while Mateus Fernandes shot straight at Pope from 20 yards.
At the other end, Hermansen saved a Lewis Hall drive moments and Barnes saw one blocked moments before Guimarães lifted a curling effort just over the bar from a similar position to where he scored a memorable goal in a win over Chelsea a year ago.
But they didn't have to wait too much longer for a third to arrive. Substitute Willock, on for Sandro Tonali, combined beautifully with Osula, who finished with aplomb.
Just moments later, however, Castellanos' stunning 30-yarder - hit first-time when a long Hermansen goal kick fell into his path - looped over a stranded Pope and made it 3-1 to gave the Hammers faint hopes of a comeback.
Wilson then saw a penalty appeal turned down after a brief VAR check triggered by a Hall challenge, and Pope spilled a Jarrod Bowen shot from the edge of the box as the Hammers showed some fight. Castellanos then latched onto another deep centre from Diouf and took aim, but the ball struck the crossbar on its way over.
Pope had to save twice more from Castellanos, showing good reactions to turn away his glancing header before clutching the rebound, and West Ham might have felt that on another day they could have taken something from a game that was so crucial to their survival hopes.
But the day belonged to the hosts and Trippier, who was withdrawn with five minutes remaining to a wonderful standing ovation in appreciation of his four-and-a-half years of sterling service on Tyneside.
Newcastle United: Nick Pope, Kieran Trippier (Anthony Elanga 86) Lewis Hall, Sven Botman, Sandro Tonali (Joe Willock 53), Harvey Barnes (Jacob Murphy 75), Malick Thiaw, Will Osula, Nick Woltemade (Dan Burn 75), Bruno Guimarães (c), Jacob Ramsey
Subs not used: Aaron Ramsdale, Yoane Wissa, Anthony Gordon, Alex Murphy, Sean Neave




