Reach's 30 yard thunderbolt after just 16 seconds paved the way for the Owls to clinch their first away victory since August 19.
Jordan Rhodes was also on the scoresheet for the second match running, grabbing a second in the 23rd minute to end Villa's unbeaten home record.
Wednesday were forced to dig deep in the second half and centre-back Chris Samba pulled one back late on for Villa.
But back-to-back victories moves the Owls into the top half of the Championship heading into the two-week international break.
Head coach Carlos Carvalhal made just one alteration to the team who defeated Millwall, recalling Jack Hunt at right-back at the expense of Liam Palmer.
There was no Kieran Lee in the matchday squad. The midfielder misses out due to a hip injury.
It took the Owls just 16 seconds to break the deadlock and it was a goal of the season contender by Reach.
Ross Wallace touched the ball into Reach's path and the left winger's spectacular 30 yard strike flew past Sam Johnstone into the top corner to stun the majority of the crowd at Villa Park. It was the 24-year-old's third goal in a week.
For our Owls news and opinion on twitter follow hereWednesday took a grip of the midfield battle, with Jacob Butterfield and Barry Bannan impressing with and without the ball.
Wallace blazed wide from long range after good work by Gary Hooper as the visitors pressed for a second.
Robert Snodgrass, who was heavily linked with a move to Hillsborough earlier this year, forced Keiren Westwood into action in the 14th minute. The Scotland international's rasping left foot drive was well saved by Westwood.
Soon after Bannan, on his return to Villa Park, sliced well wide after lovely approach play on the right flank involving Jack Hunt, Wallace and Reach.
Villa suffered a blow midway through the first half as John Terry limped off with a knock and was replaced by Chris Samba.
And the Owls exploited the hosts defensive uncertainty in the 23rd minute when Rhodes doubled their lead. He fired home after latching on to Hooper's pass following a surging run by Bannan.
The scorers then combined again, with Reach drilling a shot well wide.
Villa looked disjointed and short on ideas in the final third. Samba headed straight at Westwood in one of their rare fluid attacks.
Their limp performance prompted manager Steve Bruce to make a tactical change before the interval. He took off Robert Snodgrass, who threw his shin pad down in disgust at the decision, and Callum O'Hare was introduced to midfield.
For more news from Hillsborough click hereWhatever Bruce said at half-time had the desired affect as Villa started the second period on the front foot. It took a last-ditch challenge by Tom Lees to stop Scott Hogan testing the reflexes of Westwood.
Wednesday struggled to get out of their own half, with Villa dominating possession.
Wallace summed up their frustration, receiving the first yellow card of the afternoon after scything down Keinan Davis.
The Owls created their first meaningful opportunity of the second half on the hour mark, Wallace shooting high over the top after a cushioned header by Rhodes. Injury forced Wallace off and he was replaced by Liam Palmer, who filled in on the right flank.
With 11 minutes remaining, Carvalhal's men almost increased their advantage when Hooper slipped in Reach and his neat pull back was turned wide by Palmer.
Villa had a goal disallowed for offside in the dying minutes and James Chester headed over Conor Hourihane's free-kick as they continued to threaten.
And Samba bagged a late consolation, tapping in O'Hare's centre but it proved too little, too late.
Villa: Johnstone; Elmohamady, Chester, Terry (Samba 21), Hutton; Snodgrass (O'Hare 40), Hourihane, Whelan (Grealish 77), Adomah; Davis, Hogan. Substitutes: Steer, Taylor, Jedinak, Bjarnason.
Owls: Westwood; Hunt, Lees, Loovens, Fox; Wallace (Palmer 63), Butterfield, Bannan, Reach; Hooper (Jones 83), Rhodes (Fletcher 73). Substitutes: Wildsmith, van Aken, Joao, Abdi.
Referee: Geoff Eltringham (County Durham)