With Dele Alli on the bench and Harry Kane given the day off – two of nine changes to the Tottenham side who beat Premier League leaders Chelsea last weekend – the hosts struggled for goalscoring opportunities for long periods at White Hart Lane.
After a first half in which they failed to get a shot on target, it was left to an unlikely source to finally break the deadlock inside the final 20 minutes.
Davies had not scored since netting for Swansea City against Arsenal September 2013, but demonstrated his contortion skills to head home on his 69th Spurs appearance.
And Son Heung-min made sure of the win when he fired in from 12 yards 10 minutes from time.
The win is Tottenham's 13th in their last 17 matches against Villa, having lost just one.
A Jack Grealish free-kick into the box gave the home defence some work to do three minutes later after stand-in skipper Eric Dier clumsily brought down Bacuna, while Tottenham continued to struggle to carve open the Villa back line.
A lack of Tottenham bodies in the penalty area – perhaps a side-effect of the number of changes – spelt the end of attacking moves on more than one occasion, although Villa full-back Jordan Amavi almost chipped Johnstone with a mishit clearance that would have gifted the home side a half-time lead.
Vincent Janssen and Son exchanged passes on the edge of the box before the Dutchman mustered Spurs' first shot on target early in the second half – a well-struck effort that was comfortably held.
Son was then unable to make the most of a Kieran Trippier cross from the right that found him in space in the box and it was no surprise to see Alli introduced on the hour mark.
The addition of Alli in place of Janssen saw Son move to the head of Spurs' attack and the South Korea international drew a superb diving save from Johnstone five minutes later.
Substitute Georges-Kevin Nkoudou's cross was delivered behind Davies, but the Wales international adjusted his body superbly to divert beyond Johnstone.
And victory was in the bag when Son was left with a straightforward finish from the centre of the area after Moussa Sissoko's cutback.