They were locked at 4-4 with holders Auchinleck Talbot after normal time, but the Irvine outfit prevailed 4-2 in the penalty shoot-out, with Mark McLelland stroking home the decisive spot-kick after Talbot pair Ray Montgomerie and Bryan Young missed with their efforts.
Talbot had raced into a fourth-minute lead after Keir Millican rounded the keeper, before a fortuitous break of the ball led to Liam McGuinesss levelling and then Robert Halliday blasted Medda ahead.
Auchinleck restored parity straight from the restart through Kyle Faulds, but before half-time goalscorer Milliken was sent packing for two yellow card offences inside three minutes.
The 10 men belied their numerical disadvantage to go 3-2 up in 52 minutes through a Willie Kinniburgh own goal, but Frank Gibson spectacularly levelled with an overhead kick.
Unbelievably, Auchinleck went straight upfield and Gordon Pope made it 4-3 from a penalty kick.
A last gasp foray saw Tom Brighton touch home for an amazing 4-4 scoreline and then came Medda's dramatic spot-kick triumph.
Victorious boss Strain said: "The players never gave up the fight. Some of them were only half-fit yet our injury situation meant we had to ask them to give their all ... and what a response they gave."