Dawson's accurate set shot from about 35 metres out has delivered the Crows a first victory of the season, 15.6 (96) to 13.14 (92).
And Port, touted as a premiership fancy, remain winless after three games after coughing up an 18-point lead some eight minutes into the final term.
The Crows booted three goals in the last five minutes to a steal a stirring victory at Adelaide Oval on Friday night.
Adelaide were two points down with just four seconds remaining when Lachlan Murphy grabbed a loose ball in the forward line and was crunched high by Port's Sam Mayes.
While given a free, Murphy was assisted off the field and unable to take his kick, with the ball given to ex-Swan Dawson.
His left-footer curled through to cap his best game for his new club - two goals and 25 disposals.
Dawson's teammates Elliott Himmelberg and Lachlan Gollant, in just his third AFL game, booted four goals each.
And Crow midfielders Harry Schoenberg (23 possessions), Matt Crouch (29), Rory Laird (31) and Ben Keays (25) were prolific ball-winners.
Port Adelaide's maligned forward Todd Marshall slotted five goals while Mayes and Georgiades each kicked two.
The Power held a seemingly commanding 18-point lead some eight minutes into the last term.
But inside five minutes remaining, Himmelberg scored two late majors before the last-gasp drama centred on Dawson.
His match-winner came after Port veteran Travis Boak missed a set-shot with one minute 40 seconds remaining, leaving the Power two points in front.
Boak (28 disposals), Ryan Burton (22), Ollie Wines (30) and Karl Amon (22) were among Port's best players.
But Ken Hinkley's side remain in the mire with three consecutive losses and a looming Thursday night fixture against reigning premiers Melbourne.
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