The opener's score with 15 sixes is also the highest and third fastest hundred for Australia as the visitors clinch the series 2-0 with one game to play.
Watson rains record sixes in unbeaten 185 to pummel Bangladesh
The explosive opener gave a superb display of power-hitting as he also cracked 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.
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Watson's knock 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 just 69 balls, the third-fastest by an Australia after Hayden (66 balls) and Gilchrist (67 balls). The final one-dayer will be played in Dhaka on Wednesday.
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 just 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.
Bangladesh were earlier struggling at 88 for five 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.
Johnson did the early damage when he removed openers Tamim Iqbal and Imrul Kayes before paceman John Hastings accounted for Raqibul Hasan to reduce the hosts to 28 for three in the opening 12 overs.
Smith bowled captain Shakib Al Hasan for nine runs and then dismissed well-set Nafees to send the hosts reeling at 88 for five.
Tamim Iqbal c Watson b Johnson 5
Imrul Kayes c Watson b Johnson 5
Shariar Nafees c & b Smith 56
Roqibul Hassan b Hastings 0
Shakib Al Hasan b Smith 9
Mushfiqur Rahim not out 81
Mahmudullah lbw b Watson 38
Sohrawardy Shuvo c Smith b Johnston 16
Shafiul Islam not out 9
Extras: (4b,1lb, 3w, 2nb) 10.
TOTAL: (for seven wickets, 50 overs) 229.
Fall of wickets: 1-9, 2-27, 3-28, 4-65, 5-88, 6-167, 7-195.
Did not bat: Rubel Hossain, Abdur Razzak.
Bowling: Bret Lee 10-2-37-0 (1nb), Mitchell Johnson 10-0-54-3 (1nb, 2w), John Hastings 6-0-20-1, Shane Watson 7-0-35-1, Steve Smith 7-0-34-2 (1w), Xavier Doherty 10-0-44-0.
Shane Watson not out 185
Brad Haddin c Mahmudullah b Shakib 8
Ricky Ponting not out 37
Extras: (1w, 1nb) 2.
TOTAL: (for one wicket, 25 overs) 232.
Fall of wickets: 1-62.
Did not bat: Michael Clarke, Callum Ferguson, Mike Hussey, Steve Smith, Mitchell Johnson, John Hastings, Brett Lee, Xavier Doherty.
Bowling: Shafiul Islam 2-0-27-0 (1w), Abdur Razzak 7-0-58-0, Shakib Al Hasan 7-1-35-1, Rubel Hossain 6-0-57-0 (1nb), Sohrawardy Shuvo 3-0-46-0, Mahmudullah 1-0-9-0.
Series: Australia leads 2-0 in the three-match series.