Davies registered his first goal for the Young Lions in the 52nd minute when he connected with a Kyle Walker-Peters cross and goalkeeper Francisco Pires failed to keep the ball out.
Having secured three points from what was a largely underwhelming outing, while Ukraine and Holland drew 1-1 and Scotland won 2-0 in Latvia, Aidy Boothroyd's England side are now four clear at the top of Group Four.
They have 10 points from four games, while the second-placed Scots have six from three.
Boothroyd made eight changes to his starting line-up from Friday's 3-1 win over Scotland, with midfielder Davies among five Everton players included in the team.
But there was not a great deal more from England in the first half, while their goalkeeper Angus Gunn was called upon to block a shot from Claudi Bove around the half-hour mark.
The only real first-half effort thereafter from the visitors was Joe Worrall's header wide, and shortly after the break Davies and Ainsley Maitland-Niles scuffed similar disappointing efforts well wide of the target.
But England were celebrating moments later when Davies nodded Walker-Peters' delivery goalwards and Pires got hands to the ball but allowed it to squirm away from him and into the corner of the net.
Albert Alavedra had a strike deflect behind as Andorra went in search of an equaliser.
The Everton contingent were then back to the fore, with Ademola Lookman sending a free-kick wide and Calvert-Lewin converting a shot from close-range, only for the effort to be ruled out with the flag up for offside.
Lookman subsequently passed up a great chance to double the advantage in the 82nd minute as his strike was saved by Pires.
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