Armstrong finished the 29km route in 37:34 to finish 15 seconds clear of the field.
World champion Judith Arndt of Germany finished in 37:50 to place second, with Russia's Olga Zabelinskaya third in 37:57 for her second bronze medal of the Games.
Great Britain's Emma Pooley, the 2010 world champion and silver medallist in Beijing four years' ago, finished in 38:37, placing sixth.
Lizzie Armitstead, Britain's first medallist of London 2012 after winning silver in Sunday's road race, clocked 39:26 to place 10th.
