England's Ashley Young and Ashley Cole missed their penalties, leaving Diamanti to coolly slot home and earn the Italians a semi-final clash with Germany on Thursday.
Italy struck the woodwork twice and had a goal disallowed in extra time while England failed to trouble Italy goalkeeper Gianluigi Buffon after an early chance for Glen Johnson.
"We played a great game and deserved to win," Italy coach Cesare Prandelli told reporters. "They haven't just been great, they've been more than that and then in penalties you always need a bit of luck."
Italy started strongly and Daniele De Rossi crashed a speculative 20-metre shot off the inside of the post in the third minute.
Johnson's scooped effort was clawed away by Buffon but Italy continued to create good opportunities, striker Mario Balotelli squandering several and kicking the post in frustration at one point.
The Italian attacking partnership of Balotelli and Antonio Cassano created problems for England's defence with clever flicks and movement, while De Rossi and Andrea Pirlo were content to shoot from distance.
England's Joe Hart made a stunning double save from De Rossi and Balotelli in the 52nd minute before Riccardo Montolivo fired over the crossbar with the goalkeeper stranded.
England winger Young fired wide in the 65th minute after substitutes Andy Carroll and Theo Walcott created an opening.
Pirlo pulled the stings for Italy throughout the match, his deft dribbles and passing often providing the platform for his side's attacks.
Carroll's aerial presence gave a new dimension to the English attack and Wayne Rooney almost connected with captain Steven Gerrard's dangerous free-kick.
Diamanti forced Hart into a smart save after coming on as substitute for Antonio Cassano with 10 minutes left and Rooney blazed an overhead kick over the crossbar in the last minute of stoppage time.
Italy continued to dominate possession in extra time and almost made the breakthrough when Diamanti's curling cross clipped the outside of the post with Hart beaten.
England's exhausted-looking side dropped even deeper in the last 15 minutes of the extra period as the Italians laid siege to Hart's goal.
Italian substitute Antonio Nocerino headed the ball into the net but the effort was ruled out for offside.
In a battle between two teams with poor records in shootouts, England had the early advantage when Montolivo dragged his shot wide.
But after Young crashed his spot-kick against the bar, Nocerino scored, Cole's effort was saved, and Diamanti was left to seal victory for Italy.
"The players have given everything," said England captain Gerrard. "I thought we might have the luck in the penalty shootout but it wasn't to be.
"When you get your nose in front during a shoot-out, you just hope you can hang on but credit to Italy, they got the luck in the shootout. The lads at the back were fantastic, we have don
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