After leaping high into the air in a bid to catch Nic White's 22nd-minute restart, Banks' left arm bent the wrong way as he crashed to the ground and attempted to avoid a head-first landing.
The Wallabies were completing their warm-up when Cooper injured his calf. Noah Lolesio was moved into the starting lineup in place of Cooper at No 10 while James O'Connor was brought onto the bench.
Cooper led the Wallabies to five straight wins last year after his four-year exile from the national side. O'Connor was part of a corporate function before the match but was rushed to the dressing room to get changed once Cooper's injury was confirmed.
Australia captain Michael Hooper said his side improvised well.
"We came up with a lot of solutions, so I'm proud of the effort despite a lot of changes at the last minute," Hooper said. "You can't script that stuff. You can have all those plans and then they go out the window. Our coaching staff did a great job of managing the game."
England captain Courtney Lawes said his side were disappointed.
"Full credit to Australia," Lawes said. "We probably didn't execute as well as we should have, and they finished their opportunities."
- AP