Newcastle kept Tottenham, who finished runners-up last term, at arm's length during a lacklustre first half, but Shelvey trod on Alli's ankle three minutes into the second period and was shown a straight red card by referee Andre Marriner.
Rafael Benitez's side, who had already lost Paul Dummett and debutant Florian Lejeune to injury, were unable to contain Spurs thereafter, with Alli making the breakthrough when he stretched to volley Christian Eriksen's exquisite delivery into the back of the net.
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.
2 - Ben Davies has scored twice in his last two @premierleague appearances - as many as in his previous 102 combined. Rosy.— OptaJoe (@OptaJoe) August 13, 2017
It will be a pleasing result for Spurs boss Mauricio Pochettino, who has fielded difficult questions following Tottenham's absence of movement in the transfer market and the explosive comments from Danny Rose in which the left-back hit out at the way the club has been operating.
Harry Kane, last season's Golden Boot winner, could have added more gloss but struck the post late on and his wait for a Premier League goal in August continues.
Benitez, meanwhile, will be hugely frustrated that Newcastle's return to the top flight was spoiled by Shelvey’s peevishness.
Dwight Gayle overcame a hamstring injury to lead the Newcastle line and sliced over the crossbar when presented with an opportunity by Dummett, who soon suffered a similar issue and was forced off after just seven minutes – Jamaal Lascelles his replacement.
Tottenham controlled possession but Gayle threatened in behind and went down in the box under contact from Kyle Walker-Peters, making his Spurs debut at right back with Kyle Walker sold to Manchester City and Kieran Trippier out with an ankle injury, only to see the flag raised for offside.
Spurs lacked ingenuity and Kane was shown a yellow card for scything down Lejeune in a tackle that brought the Frenchman's outing to an early end.
Eriksen worked Rob Elliot four minutes before half-time with a dipping, 25-yard drive the goalkeeper saw all the way and was able to hold.
There was drama just three minutes after the restart when Shelvey was dismissed for reacting to Alli kicking the ball away while down on the ground by treading on his ankle.
The game suddenly burst into life, with Gayle forcing Hugo Lloris into a save at his near post, but Newcastle's positive response soon fizzled out.
A deflection off Matt Ritchie saw Eriksen's pass loop up for the unmarked Kane inside the box, but Elliot stayed big and blocked his powerful strike in the 51st minute.
Moussa Sissoko was booed as his return to St James' Park ended with Son Heung-min taking his place before the hour mark, and Tottenham soon made their extra man count.
Alli lost Lascelles inside the box and guided a lovely ball from the right courtesy of Eriksen across Elliot and into the net.
The England international 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.
Alli was booed as he made way for Victor Wanyama in the 82nd minute and Kane was unable to rub salt into Newcastle's wounds when his fierce strike cannoned back off the upright in stoppage time, meaning his August Premier League drought extends to 11 games, but Shelvey's moment of madness had already proved the Magpies' undoing.
Key Opta stats:
- Tottenham have won 29 of their last 31 Premier League games against newly promoted sides, drawing the other two.
- Dele Alli has scored 13 Premier League goals in 2017; among English players, only Harry Kane (21) has scored more.
- Jonjo Shelvey was the first Newcastle player to be sent off on the opening day of a Premier League season at St. James’ Park since Alan Shearer in 1999.
- Shelvey has been sent off twice under Rafael Benitez at Newcastle, more than any other player. He was also red carded in a Championship match at Nottingham Forest in December 2016.
- Since the start of the 2015/16 season, Christian Eriksen has assisted more Premier League goals than any other player (30).
- Florian Lejeune became the 200th French player to feature in the Premier League and also the 27th to represent Newcastle, more than any other side in the competition’s history.
- Ben Davies has scored twice in his last two Premier League appearances, as many as in his previous 102 combined.
- None of the last 10 winners of the Championship have won their opening match in the Premier League the following season (D2 L8).