Norwich winger Redmond struck in the 56th minute, cutting in from the left flank and shooting low into the near corner of the net from outside the box.
The Young Lions went into the contest having netted 21 goals in their previous four matches, but struggled to pose an attacking threat in a subdued first half and might even have gone in at half-time 1-0 down.
It took 25 minutes for the first real attempt to arrive, with Ravel Morrison drilling wide, and five minutes later England survived a scare when Wes Burns broke clear at the other end, only to hit his shot across the face of Jack Butland's goal.
Gareth Southgate's side continued to generally control possession thereafter but kept sending efforts off-target, with the likes of Saido Berahino, Redmond and Tom Ince trying their luck unsuccessfully.
England looked a little more menacing as the second half got under way and after warning shots from Ince and Redmond, the latter broke the deadlock in a decent individual moment as he danced through a clutch of Wales players before squeezing the ball past goalkeeper Connor Roberts.
The home side tried to keep the tempo up, but in the end Redmond's effort proved sufficient for a victory that adds to their unbeaten record in Group 1.
In their last match, Southgate's team - albeit against San Marino at home - had raced into a 4-0 lead inside the opening 25 minutes en route to a comprehensive 9-0 victory.
They barely registered anything that could be considered a shot within the same time period at the start of this game, before Morrison finally broke the pattern with his low drive wide.
The Welsh will have been content with how things were proceeding but might have felt disappointed that Burns could not make more of his chance on the half-hour.
England's response saw James Ward-Prowse put a speculative effort wide from distance, Berahino's cheeky flick in the box deflect behind and Redmond cut inside and send a shot off-target.
Ince then broke into the area but scooped over, before Redmond headed clear of the crossbar in a tepid conclusion to a forgettable half.
Wales defender Joe Walsh almost diverted a cross into his own net shortly after the restart, before Tom Lawrence curled a shot from the edge of the area wide of Butland's post.
But England then began to step things up, with Ince firing a shot with some venom too high and Redmond bringing a dive out of Roberts with an attempt that bounced just the wrong side of the post.
And two minutes later, the pressure told as Redmond escaped several red shirts and beat Roberts, notching his second goal in nine under-21 caps.
Berahino headed over and Ince put the ball on the roof of the net as England went in search of a second, before Derby starlet Will Hughes came on as a substitute, receiving a warm reception from the crowd.
Tom Carroll saw his shot deflect behind and another England substitute, Danny Ings, had a late strike pushed away by Roberts, but England had done enough.
Southgate's men, top of the six teams in the qualifying pool, now have 19 points from seven games, while Wales have 10 from seven.