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.
Gayle forced a near-post save from Hugo Lloris in the immediate aftermath, but opposite number Elliot had to make a fine 52nd-minute block to deny Harry Kane the opening goal.
Christian Atsu and Javier Manquillo impress Newcastle United fans despite Tottenham loss
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.
The Toon midfielder was given his marching orders in the second half of the Premier League clash on Sunday after leaving one on Alli, causing widespread confusion over his freakish decision. Christian Eriksen and Dele Alli support Harry Kane.
And to nobody's surprise - except Shelvey's - Mariner brandished a straight red card. Eric Dier plays, as does Moussa Dembele, while there is also a start for former Magpie Moussa Sissoko.
The brain-fade derailed Newcastle's momentum and any chance of pulling off an upset against last years Premier League runner-ups.
Troopers identified as two who died in helicopter crash linked to rally
A suspect, identified as 20-year-old James Fields from OH , has been detained on suspicion of second-degree murder. National Transportation Safety Board spokesman Peter Knudson said the cause of the crash had not been determined.