Victory, achieved with more than two days to spare, saw England go 2-1 up in the five-match Ashes series.
England, set a modest victory target of 121, finished on 124 for two after lunch on Friday's third day.
Bell was 65 not out and Joe Root, who struck the winning boundary off Mitchell Marsh, 38 not out.
There was a moment of concern for England when captain Alastair Cook (seven) was bowled by a superb swinging delivery from left-arm paceman Mitchell Starc to leave the hosts seven for one.
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Bell, arguably England's most elegant batsman, then late cut off-spinner Nathan Lyon to the third man rope.
He was, however, given a reprieve on 20 when, with England 35 for one, he edged Starc to second slip only for Australia captain Michael Clarke to drop the two-handed catch.
It summed up a miserable match for Clarke, whose run of low scores in Tests continued with innings of 10 and three at Edgbaston.
Root, Lyth's Yorkshire team-mate, showed few signs of nerves, however, in cutting his first ball, from Hazlewood, for four.
Bell's late-cut boundary off Starc saw him to his second fifty of the match, off 68 balls with seven fours.
It was a fine performance by the 33-year-old Bell, who came into this match -- his 113th Test -- with a question mark against his place after several meagre innings.
It appeared to get to the left-arm quick, who aborted his run-up for one delivery and then bowled from well behind the crease before he was taken off.