After a goalless first half at the Tottenham Hotspur Stadium, Elanga's swivelled finish fired the visitors in front shortly after the hour mark.
Wissa's improvised finish, his first competitive goal since January, doubled United's advantage as Matthias Jaissle recorded his first Premier League win in charge of the Magpies.
The German head coach made six changes from Wednesday's 3-2 win against West Bromwich Albion in the Carabao Cup, with Nico González making his Magpies bow following his transfer from Manchester City.
Targeting a third successive away win against Spurs, Newcastle went close to taking the lead inside the opening 60 seconds.
Elanga, who netted his first Premier League goal for Newcastle in last weekend's 2-2 draw against Liverpool, latched onto Amar Dedić's clever header following Nico González's long-range pass but the Swedish winger's cross-cum shot was turned out by Archie Gray at the back post.
Lukáš Horníček produced a strong stop to prevent Mathys Tel from breaking the deadlock as the French forward cut inside from the left flank before his curling effort was pushed away by the Czech shot-stopper.
Lewis Hall produced a heroic last-ditch block to prevent Pedro Porro from firing Spurs ahead minutes later, turning the Spanish full-back's close-range effort over the crossbar.
The Spurs full-back went close in front of goal as the Magpies defence were unable to clear from a short corner before his volley from the edge of the box flew wide of the target.
Wissa went close to netting his first Premier League goal since December as the DR Congo international attempted to fire a shot at goal inside the six-yard area after Elanga nodded the ball into the African striker's path but his effort was blocked by Micky van de Ven.
Horníček carried out two terrific saves before the half-time whistle, beating away Andy Robertson's low drive from an acute angle. From the resulting corner, the Czech 'keeper was alert to pushing away Jan Paul van Hecke's close-range header.
Lewis Hall went close to breaking the deadlock shortly after the restart as the left-back cut inside on the edge of the box before his curling right-footed effort flew over the crossbar.
In the 62nd minute, Elanga broke the deadlock for United. After Nico's impressive long-range pass flew into Dedić's path, the Bosnian full-back nodded the ball into Elanga's path before the Swedish frontman produced a neat turn before firing the ball into the net via the left-hand post.
Horníček prevented Porro from drawing Spurs level as the Spaniard's fierce drive from the edge of the box was tipped over the crossbar by the 24-year-old 'keeper.
In the 72nd minute, Newcastle extended their advantage as Wissa netted his first goal of the 2026/27 campaign. After Hall's inviting corner from the right flank was nodded towards substitute Nick Woltemade, Wissa was on hand to turn the ball home from close-range.
Horníček ensured a first clean sheet of his Magpies career in the closing stages, holding Omar Marmoush's volley after Mateus Fernandes' cross found the unmarked Egyptian inside the penalty area.
There was a brief Premier League debut for Bazoumana Touré as the Ivorian winger, who scored on his competitive Magpies bow during Wednesday's win against West Brom, was introduced in stoppage-time.
Newcastle United: Lukáš Horníček, Lewis Hall, Sven Botman, Nico González (Sean Steur 71), Yoane Wissa (Jacob Ramsey 82), Harvey Barnes (Bazoumana Touré 90), Malick Thiaw (c), Anthony Elanga (Jacob Murphy 82), Joe Willock (Nick Woltemade 71), Amar Dedić, Lewis Miley.
Subs not used: Ewen Jaouen, Fabian Schär, Aladji Bamba, Leo Shahar.
