The news will be encouraging to the music industry, which is returning to growthafter piracy brought it to its knees in the 2000s.
It has campaigned heavily to shut down sites that offer pirated music - from Napster and allofmp3 to Pirate Bay and Megaupload.
Among those surveyed by YouGov, 36% said it was becoming more difficult to find unauthorised sources to verify music.
However, record labels remain concerned by the rise of "stream ripping" services.
Such apps allow users to "grab" music from streaming services like YouTube and Spotify and store the file on their phone or computer, which prevents artists and record labels getting revenue from future streams.
YouGov noted that some illegal downloading may be the result of streaming exclusives - with 44% of respondents saying they only download songs illegally when they can't access them elsewhere.
One of this year's most-pirated albums is Beyonce and Jay-Z's Everything Is Love, which was initially released as an exclusive on Tidal.
Justin Marshall, associate director of YouGov, said: "While illegal downloads still present a significant challenge to the music industry, there appears to be some light at the end of the tunnel.
"Whether or not streaming is what finally banishes illegal downloads remains to be seen, but there are encouraging signs."
YouGov surveyed 4,009 UK adults between 6-13 March this year.
However, its findings may have been skewed by respondents who were reluctant to admit accessing music illegally.