The Magpies were playing their part in a close battle but their game unfolded when Jonjo Shelvey was shown a red card in the 48th minute.
Alli and Davies then pounced before full-time as Tottenham opened their Premier League campaign with a win away from home.
Newcastle had the first chance of the match when Dwight Gayle was played in over the top but he could only volley the ball over the bar in the fifth minute.
Manager Rafa Benitez was then forced to make two changes as Paul Dummett and Florian Lejeune were forced off with injuries.
Neither side took control of the first half but the momentum swung in Spurs' favour when Shelvey was sent off inside the first three minutes of the second period.
He appeared to stamp on Alli's foot after a challenge and his dismissal changed the game for both sides.
Harry Kane missed a one-on-one chance in the 51st minute but Alli opened the scoring 10 minutes later. His run was picked out by Christian Eriksen and the midfield duo combined to put Spurs ahead.
Eriksen was involved for Davies' strike as his lay-off teed up the left-back and the Wales international curled the ball into the far corner.
In stoppage time Kane hit the post and had a goal disallowed for offside but Spurs had already done enough as they capitalised on another moment of madness from Shelvey.