Smith's knock of 215 is the second highest Australian score at the home of cricket, bettered only by Bradman's 254 in 1930.
The 26-year-old is also the first Australian since Bradman in 1934 to make a 200-plus score and a 50-plus score in an Ashes Test.
His Test average is now 58.52.
That figure is topped by only one Australian to have played more than 15 Tests - Bradman.
In the 20 out of 30 Tests that he has played, Smith averages 96.61 as compared to Bradman's 113.66 when Australia have batted first.
Smith's incredible purple patch shows no sign of stopping, but the gifted right-hander has plenty of work to do before comparisons extend beyond the statistical realm and into reality.
Named man of the match on Sunday, Smith was simply happy to bank a big score after missing out in the first Test.
"We were all disappointed with the way we performed with the bat in Cardiff," Smith said.
"I was pretty keen to do well here this time around, I haven't had such a great run with the bat at Lord's."
Smith suggested he had to tweak his approach in England.
"The wickets are generally a little bit slower than they are back home," he said.
"You've got to wait for the ball a lot more. The ball does a lot more than it does back home.
"You need to be really patient, it's about letting balls go."
That may have been the case during Smith's 512-minute masterclass on days one and two, but Sunday was more about hit and giggle.
Smith embraced his inner funkmeister at the crease, shuffling around the crease and making good use of trick shots to score 58 off 48 balls.
"It was a little bit of fun," Smith said of the knock.
Michael Clarke paid tribute to Smith and Chris Rogers, who shared a record 284-run partnership.
"They deserve a lot of credit in halting the momentum that England had from the first Test," Clarke said.
- AAP
