Nico Yennaris fired the Bees into an early lead, but Gary Hooper equalised in first-half stoppage time before Wallace smashed home the winner 20 minutes from time.
But Dalsgaard went down injured deep into added time and eventually left the pitch on a stretcher after receiving treatment on the pitch for over 10 minutes.
The visitors stung the Owls after just nine minutes thanks to a goalkeeping error from Keiren Westwood.
A Yoann Barbet free-kick had flicked off the wall and behind for a corner and from the resultant set-piece the ball was cleared to Yennaris, who hit a first-time shot from the edge of the area. Westwood looked to have saved the effort but he only proceeded to fumble the ball into his own net.
Wednesday almost hit back immediately, with Wallace's terrific run and shot bringing an excellent save from Daniel Bentley. The Bees keeper then produced an even better stop from Tom Lees' header, low to his left, after Wallace whipped in the corner.
Westwood then had to be alert to tip away the unmarked Neal Maupay's shot after the Wednesday defence froze, expecting the Frenchman to be flagged offside.
Brentford were producing some exquisite passing football and Josh Clarke had their next effort, but his tame shot was easily gathered by Westwood.
But Wednesday drew level before the break.
Adam Reach took the ball off Ollie Watkins' toes on the left flank and played a delightful through ball to Hooper, who shrugged burst into the box and dinked the ball over Bentley. John Egan got back to knock his effort against a post, but Hooper followed up to slot the ball home.
Yennaris tested Westwood from range within two minutes of the second half, before Barry Bannan forced a save from Bentley at the other end, with the visiting keeper then thwarting Kieran Lee.
Bannan's shot was deflected wide and just like Brentford in the first half, the resultant 70th-minute corner produced a goal as Wallace smashed the ball into the net after Rhodes headed back across the area.
Brentford substitute Florian Jozefzoon should have done better when played in on goal, with his shot almost going out for a throw-in as the game entered its last 10 minutes, before Lees blocked Maupay's effort as the visitors were ultimately denied.
There was concern around the stadium as Dalsgaard went down injured and remained motionless for several minutes as medics treated him on the pitch, before stretchering him off.
The Bees almost levelled in the 17th minute of added time but Maupay fired wide from the final play of the match.