A flurry of punches was thrown when Bradshaw retaliated after being head- butted by Rush. Both managers, Denis Smith and Mike Walker, rushed from the dug-out but were too late to prevent the brawl and the referee, Alan Wiley, showed the players the red card.
"Bradshaw's a bit fiery and I thought it best to intervene but you cannot condone retaliation and the referee had no choice," Walker said.
The Canaries' victory sets up next week's second leg of the Coca-Cola Cup between the two teams which is evenly balanced at 1-1.
A goal from Neil Adams in the 12th minute was enough to separate the two teams. Last season it would have been doubtful if Norwich would have been able to survive the test provided by Oxford's battlers, but the Canaries' new-found resilience proved too much for Denis Smith's team.
After their 5-0 win over the weekend Oxford were full of confidence as they continually attacked Norwich's goal. But they were eventually worn down by the visitors.
The return of Walker after his ill-fated period at Everton, has added an extra weapon to Norwich's armoury as he aims to repeat the success of his previous spell at Carrow Road.
Walker maybe a long way from Europe at the moment but in the first half hour Norwich showed glimpses of the form that beat Bayern Munich.
Intricate passing and determined battling brought Neil Adams' deciding goal. The wellworked move began when Ian Crook turned his marker and provided the charging Danny Mills on the left wing with a precise pass which took out two Oxford players.
Mills then crossed for Adams whose header sneaked to the left of a despairing Phil Whitehead in the Oxford goal.
However, Mike Walker's team will have to produce this standard of football for more than than just 30 minutes if they are to return to the Premiership.
In the second half Oxford could only managed to attack in fits and starts. The closest they came was when Matt Elliott received a defence-splitting pass from Bobby Ford and played the ball into the path of Nigel Jemson and the onrushing Norwich goalkeeper Bryan Gunn.
Jemson was injured in the clash between the two and was taken off by Smith. Norwich were then unable to find their intricate and free-flowing form of the first half, but this was still enough to keep Oxford at bay.
Oxford United (4-4-2): Whitehead; Robinson, Elliott, Gilchrist, M Ford; B Ford, Smith, Massey (Gray, 76), Beauchamp; Moody, Jemson (Rush, 67). Substitute not used: Aldridge.
Norwich City (1-3-5-1): Gunn; Newman; Bradshaw, Polston, Mills; Adams, Crook, Johnson, Milligan, Eadie; Fleck (Akinbiyi, 79). Substitutes not used: Carey, Marshall.
Referee: A Wiley (Burntwood, Staffs).Reuse content