Sarri's side began brightly despite the absence of Eden Hazard, who started on the substitutes bench.
Pedro and Willian both carried a threat, but Gonzalo Higuain missed the pick of their chances from close range before the contest fizzled out.
Leicester - who have enjoyed a renaissance since Brendan Rodgers took charge in February, collecting 17 points over the last nine games - end the season in ninth.
"We were in trouble a couple of weeks ago which shows the league is not easy but we showed character in finishing third and reaching the final of the Europa League."
On Eden Hazard's future, he added: "He has been an important part of the club for the last seven years. Everyone here loves him and we would like him to stay with us."