Despite being outplayed by the Italians for long periods, Scottish pressure paid dividends as they recovered from a six-point deficit to conjure up a try before the final hooter and hand Greig Laidlaw the chance to win the match with the conversion.
It was sheer determination that created the score, winning a penalty after the 80 minutes was up and then creating space for flanker Strokosch to round the despairing tackle of Lorenzo Cittadini and go over the line.
Last year, Laidlaw did the same thing to Samoa when he slotted the winning points, and there was never any doubt he would forgo the chance to repeat the feat.
And as the ball sailed between the posts, the Scots began their celebrations, leaving Italy to wonder how it had gone wrong for them.
Scotland were given a timely boost before the match when number eight Johnnie Beattie came through a late fitness test and was able to start, but the danger Italy posed was soon on show as his side got off to the worst possible start, conceding a try inside the first minute.
The Scots failed to deal with the kick off to put themselves under pressure, before wing Tommy Seymour came rushing out of his line to leave the defence exposed.
Italian wing Leonardo Sarto used the space to wrong-foot Tom Heathcote and score an easy try, converted by Alberto Di Bernardo.
The Scots hit back in style, forcing Italy to concede a lineout on their own 22. Flanker David Denton went on a rumble through the Azzurri defence and his sublime offload sent centre Matt Scott jinking the rest of the way to the line.
With Laidlaw adding the simple conversion, the sides were level again.
A collapsed scrum and Di Bernardo's resulting penalty edged Italy ahead once more but they quickly conceded a soft score to hand the initiative back to the Scots.
Italy seemed to have cleared up a dangerous chip from Scotland cleanly enough, but Giovanbattista Venditti tried to run the ball clear and when he lost hold of it Laidlaw pounced to put wing Sean Lamont over.
The conversion and a penalty soon afterwards gave the Scots a seven-point cushion but they were living dangerously, the most obvious let-offs when Sergio Parisse missed his support player on the outside and when Sarto made another clean break down the touchline but his inside pass missed centre Alberto Sgarbi.
The move did force the Scots to carry over their own line and the Italians took full advantage, forcing their opponents to collapse a succession of scrums and inevitably winning a penalty try, which Di Bernardo converted to level the scores.
An exchange of penalties between Laidlaw and Di Bernardo before and after the break left the contest all square.
Italy then went to their bench in the search for the inspiration that could allow them to steal the game, but it was Scotland who came closest to a score with Scott again crossing the line, only for the television match official to rule that the final offload from Tim Visser had gone forward.
The near-miss lifted Italy and Scotland celebrations soon turned to grim faces as a penalty at the resulting scrum was booted clear before another kick from the resulting maul handed Di Bernardo the chance to put his side ahead.
He took it gratefully and repeated the feat a few minutes later to give his side a six-point cushion.
Scotland had a chance to haul themselves back, but Duncan Taylor was inches short and then Laidlaw knocked on as he tried to nip over the resulting ruck on the Italian line.
Another scrum and another penalty followed before Italy cleared their line.
However, all that was merely a prelude to the dramatic finish which produced Strokosch's converted try and a lucky escape for Scotland.