Redmond opened the scoring after 18 minutes when he dispossessed Lloyd Isgrove and cut in from the left before finishing coolly.
Gwion Edwards headed in a corner just two minutes later to draw Wales level, but Redmond struck again on 38 minutes with a low shot from 20 yards.
Redmond then fired England to a seventh win in eight group games to all-but book their place in the play-offs with a powerful effort just before the final whistle, although his claims for a hat-trick may have to be re-examined as the final goal just appeared to deflect in off team-mate Saido Berahino.
England made a bright start, controlling possession and territory to create chances in the opening 10 minutes for Tom Ince, Danny Ings, Jesse Lingard and Redmond as Wales received ominous early warning signs about the quality of Gareth Southgate's team.
An immaculate playing surface was made even slicker by occasional showers and that perfectly suited England's impressive passing game, with Wales finding themselves stretched on both flanks.
And the deadlock was broken after 18 minutes when Redmond ran a retreating Wales defence ragged, cutting inside from a wide position and shooting past Wales goalkeeper Connor Roberts at his near post.
But Wales responded superbly and, after Bristol City striker Wes Burns saw a shot saved by England captain Jack Butland, the home side drew level just two minutes after Redmond's strike when Edwards pounced from a corner.
The Swansea player rose high above England defenders to power a header beyond Butland as the visitors conceded only a second goal from eight starts in their European qualifying pool.
England were unquestionably rattled by Wales' recovery and Butland was soon called upon to make another save after Isgrove homed in on goal.
Both sides put a clear emphasis on attack, with England going close to regaining the lead through Ings and Lingard, but they were denied by Roberts, who then turned away a powerful Luke Garbutt shot.
Redmond, though, proved England's inspiration again seven minutes before half-time when he restored their advantage after firing past Roberts from outside the box.
The second half, much of it accompanied by torrential downpours, saw England tighten their grip as Roberts was kept busy stopping a number of short and long-range efforts, while substitute Berahino slid a good chance wide.
Wales offered little in comparison, often being outmuscled and overpowered in midfield, where Lingard proved a dominant force, although Southgate would have been disappointed that his players did not convert more than two of the numerous chances they created.
It was a relatively quiet night in defence, though, with Butland not troubled during a second period when his only worthwhile piece of action was to collect a harmless Wales corner, and Redmond had the final say with a goal that confirmed another three-point haul.