Top scorer Jon Parkin scored for the third successive match to settle the contest in the 26th minute, looping a header in at the far post and over United keeper Michael Poke following a superb deep cross from left-back Dean Howell.
Howell also cleared a Kevin Nicholson shot off the line as United pressed for a leveller, with Davies making a string of superb stops to deny Billy Bodin, Ryan Jarvis and Nicholson.
In stoppage time home central defender Aaron Downes crashed a header off the bar from Nicholson's free kick.
Alexander said: "Scott has had to earn his money and that is what he is there for.
"It was pleasing for him and pleasing for us as a team to keep another clean sheet.
"We are ecstatic because we have come down to a tough place after a monster journey.
"It was great to get the result as Torquay put us under pressure.
"We scored a very good goal and we should have had a second - I didn't see anything wrong with Jean-Michel Fontaine's header which was ruled out.
"We should have been 2-0 up at that stage but to come away and win 1-0 keeps us on the right road."