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Uruguayan Nunez put the seal on a dominant Liverpool display with his second goal in stoppage time.
Leicester were in command after Jamie Vardy's scrappy first-half goal. But with the relegation trapdoor creaking underneath their feet, Bournemouth turned the match around in stunning fashion with Dominic Solanke scoring twice.
After Liverpool's 44-match unbeaten league run ended in shocking fashion at Watford last week, victory restored their 25-point lead over Manchester City who have two games in hand.
Bournemouth profited from first-half goals by Nathan Ake and Harry Wilson, but had to fight off a home resurgence after the break before wrapping up the points with a Callum Wilson goal to end a run of 15 previous games without a win at Saints.
Results 1-4
