A tepid contest devoid of goalmouth action in the opening half sprung to life after Shelvey was dismissed for a petulant stomp on Alli, who pulled all the strings for last season's Premier League runners-up.
As for Spurs, Kyle Walker-Peters makes his top-flight debut as he solves Pochettino's crisis at right-back.
Davies added Tottenham's second with 20 minutes remaining, finishing a slick move with a composed finish beyond Newcastle goalkeeper Rob Elliot, Eriksen again the provider.
The England worldwide was involved in Tottenham's second as his flick was guided into the path of Davies by Eriksen, the left-back finishing a slick move with a cool finish.
Arsenal beat Leicester City 4-3 in Premier League opener
Having been promoted as runners-up in the Championship last season, Albion are among many pundits picks to be relegated. One of which was against West Ham, United's opponents on Sunday as the new Premier League season gets underway.
Benitez admitted that Shelvey made a mistake in standing on the ankle of Dele Alli, which caused his sending off, but the Spaniard said Harry Kane's tackle from behind on Lejuene was worse.
Spurs have now won 13 of their past 14 Premier League games, winning each of their past four and scoring 17 goals in the process.
- Shelvey has been sent off twice under Rafael Benitez at Newcastle, more than any other player. It was a clear red card [for Shelvey].
Not what you want when you've just been promoted to the Premier League.