Wilson scored 15.466 which put him in second behind gold medallist Favian Hambuechen throughout the competition.
The 20-year-old looked set to win a silver until American Danell Leyva pipped him in the last routine of the final by 0.033.
German Hambuechen set the pace from the first routine to win a first Olympic gold of his career with a score of 15.766.
Wilson had qualified for the final in second place and went on to put the disappointment of narrowly missing out on a team bronze last week firmly behind him.
The City of Leeds Gymnastics Club member also finished eighth in the men's all-around final where teammate and two-time gold medallist Max Whitlock won bronze.
Great Britain now has six medals in the gymnastics event following 16-year-old Amy Tinkler's third place in the women's floor.