Silver went to Ondrej Synek of the Czech Republic, who finished the final with a time of 6:59.37.
Drysdale finished in 6:57.82 and Northern Ireland's Alan Campbell won bronze for Great Britain with 7:03.28.
'It's a moment that will live with me forever,' Drysdale said after being hoisted on to the shoulders of Synek and Campbell.
'I have never had nerves like this before. Two hours before the race I threw up. It was one of the worst mornings of my life but when I got on the starting block I started to enjoy myself.
'It was a dream 12 years ago to be an Olympic champion. It has taken me three Olympics to get it but this is the realisation of that dream.'
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