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Italy made a bright start to the match, as some fast-paced play resulted in two tries for Aiden Guerra in the opening period, but replies from Ben Evans and Rhodri Lloyd left the host nation just two points down at the break.
Elliot Kear gave Wales the lead for the first time at the start of the second half, hauling in a cross-field kick to ground the ball in the corner.
Italy were able to up their game late in the match as they tried to seal the win, resulting in tried for Tedesco and Chris Centrone, leaving Wales needing two tries inside the last nine minutes.
Wales had to defend hard in the final minutes as the Azzuri looked to increase their score, and they found a gap in the dying seconds, allowing Mark Minichiello to make it six tries for the Italians.