After a goalless first half, Newcastle took the lead shortly after the hour mark as Nico González's diagonal pass from distance was nodded into Elanga's path by Amar Dedić before the Swedish winger placed a low effort into the net via the left-hand post.
Wissa's improvised finish extended United's advantage further, latching onto Nick Woltemade's flick following Lewis Hall's teasing corner.
The Magpies went close to scoring inside the opening 60 seconds as Nico's long-range pass was cleverly headed into Elanga's path by Dedić but the Swedish winger's cross-cum shot was turned out for a corner by Archie Gray.
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 goalkeeper.
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.
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.
Head coach Matthias Jaissle has made six changes from Wednesday's Carabao Cup second round win against West Bromwich Albion, with Nico making his Magpies debut following the Spanish midfielder's transfer from Manchester City.
The Magpies are unbeaten in their last five Premier League matches against Spurs and were victorious in their previous trip to the Tottenham Hotspur Stadium as goals from Malick Thiaw and Jacob Ramsey secured a 2-1 win in February.
**Newcastle United:** Lukáš Horníček, Lewis Hall, Sven Botman, Nico González (Sean Steur 71), Yoane Wissa, Harvey Barnes, Malick Thiaw (c), Anthony Elanga, Joe Willock (Nick Woltemade 71), Amar Dedić, Lewis Miley.
**Subs not used:** Ewen Jaouen, Fabian Schär, Aladji Bamba, Bazoumana Touré, Jacob Murphy, Jacob Ramsey, Leo Shahar.