Both teams profited from first-half own goals, with Owls goalkeeper Stephen Bywater's misfortune matched by Gary MacKenzie.
Robinson was sent off by referee James Linnington seven minutes from the end for an altercation with Reda Johnson on the touchline that left the Wednesday full-back floored.
Milton Keynes dominated early on and took the lead after 15 minutes when Luke Chadwick's 25-yard drive struck the underside of the crossbar and rebounded into the net off the luckless Bywater.
The Owls equalised in the 27th minute when Dons keeper David Martin flapped at Chris Sedgwick's cross and centre-back MacKenzie inadvertently bundled the ball into his own net.
The Dons threw on new loan signing Alan Smith shortly after the break, but the Newcastle man was unable to help his new team create a single chance in the second half. Chris O'Grady twice went close to winning it for the Owls before Robinson was given his marching orders late on for intervening in a scuffle between Johnson and Stephen Gleeson.