Diamanti struck after Ashley Young and Ashley Cole missed for England to take Italy through to a semi-final meeting with Germany in Warsaw on Thursday.
Italy had much the best of the first game of the knockout-stage to go to extra time, having more than 30 attempts at goal. Daniele De Rossi, with a swerving 25-metre shot after three minutes, and Diamanti both hit the post.
Riccardo Montolivo, twice, and Mario Balotelli, several times, also went close while Antonio Nocerino had a 114th minute headed goal correctly ruled out for offside as England's defence was continually cut open after they had started well and gone close themselves through Glen Johnson and Wayne Rooney.