The Seagulls showed seven changes to the side that started the 1-1 draw with Chelsea, with David Button, Ezequiel Schelotto, Shane Duffy, Leon Balogun, Gaetan Bong, Dale Stephens and Pascal Gross all handed starts.
The visitors made six alterations to the XI who started their 1-0 loss to Hull City, with Osaze Urhoghide, Kieran Lee, Joel Pelupessy, Jacob Murray, Massimo Luongo and Steven Fletcher all starting.
The opening minutes of the game saw neither side able to fashion a clear opening, with the two goalkeepers not seriously tested.
Twenty minutes in, the hosts finally went close as Gaetan Bong met a Steven Alzate cross in the area, but saw his header tipped away from danger by Owls keeper Cameron Dawson.
Just past the half-hour mark Albion threatened again, with Dawson making a decent block with his legs to keep out a Neal Maupay drive.
Right on half-time the Owls almost struck against the general run of play as Fletcher cracked in an effort which drew a fine one-handed save from home keeper Button.
Fletcher was stretched off early on in the second half with what looked like a knee injury, while soon after Shane Duffy saw his header from a Pascal Gross corner cleared from close to the goal-line.
The deadlock was finally broken 25 minutes from time as a free-kick just outside the area was rifled home by Reach with the help of a generous deflection to beat Button.
Maupay and Duffy came closest to rescuing something for the Premier League hosts, but neither was able to hit the target as the visitors held out to secure a hard-earned victory.