The explosive opener gave a superb display of power-hitting as he also smote 15 fours in his 96-ball knock to help Australia achieve a 230-run target with 24 overs to spare for an unbeatable 2-0 lead.
West Indies' Xavier Marshall was the previous record-holder with 12 sixes against Canada in 2008.
Watson also scored the most runs (150) in boundaries in a one-day innings, breaking South African Herschelle Gibbs's record of 126.
"It was a nice day today as everything came off," said Watson, named man of the match.
"I had no idea (of sixes record). I was pretty tired and didn't want to run too much. It was pretty hot and humid, so I tried to limit my running and it worked out nicely."
Watson's innings was also the highest by an Australian in one-day internationals, the previous best being 181 not out by Matthew Hayden.
He completed his sixth one-day hundred off 69 balls, the third-fastest by an Australian after Hayden (66 balls) and Adam Gilchrist (67).
"That's probably the best innings I've seen. He (Watson) hit the ball so cleanly from the start today," said Australia captain Michael Clarke.
Watson looked in excellent touch from the beginning, hammering three fours in the opening over bowled by Shafiul Islam. He raced to his half-century off 26 balls with the help of four sixes and five boundaries.
He then reduced Bangladesh's bowlers and fielders to a state of helplessness with his onslaught, once hitting four sixes in an over from left-arm spinner Suhrawadi Shuvo.
Watson dominated an unfinished 170-run stand for the second wicket with former captain Ricky Ponting, who contributed an unbeaten 37.
"He (Watson) batted really well and the credit goes to him," said Bangladesh captain Shakib Al Hasan.
Bangladesh were earlier struggling at 88-5 before crossing the 200 mark, thanks to wicketkeeper-batsman Mushfiqur Rahim's unbeaten 81 off 80 balls which contained one six and nine fours.
Rahim's best came in the closing stages of the innings when he smashed two fours and a six off successive deliveries from paceman Brett Lee before hitting two boundaries in seamer Mitchell Johnson's last over.
He played a major role in steadying the innings after Johnson (3-54) and leg-spinner Steven Smith (2-34) had put the hosts in trouble, adding 79 for the sixth wicket with Mohammad Mahmudullah (38).
Shahriar Nafees was the only top-order batsman to offer resistance, scoring 56 off 73 balls with six fours. He fell after completing his 12th half-century in one-dayers, offering a simple return catch to Smith.
