Saido Berahino scored the only goal of a game that England dominated from the start.
The West Bromwich Albion youngster then scored his goal, again turning his marker inside the box, but this time smashing the ball past the goalkeeper at the near post.
England wasted several more chances before the break, on top of a decent penalty shout from Wilfried Zaha, and went in one ahead.
The Moldova goalkeeper was again in action at the start of the second half, beating away powerful efforts from Zaha and Redmond.
Zaha then missed the best chance of the game, somehow heading over after Redmond's whipped delivery, before Berahino missed and even better chance. The goalscorer rounded the keeper but missed the open goal and allowed the defence to clear the ball.
Substitute Will Hughes missed a good chance late on, but England were comfortable throughout and took the three points in their Under-21 European Championship qualifier.