Skipper Swanson led the way as the Eagles held off a last-gasp Power surge to cling on for a 6.3 (39) to 4.9 (33) win in a fiery clash at Mineral Resources Park on Friday.
With only one point separating the teams at the final change, the Eagles' forward Kate Bartlett then booted two majors in her return match to give the hosts some breathing space.
But the Eagles (1-4) had to withstand a late challenge from the Power after Gemma Houghton kicked her second goal with three minutes remaining until the visitors ran out of time.
The win snaps the Eagles' 10-match losing streak and lifts the side off the bottom of the ladder as they look to avoid a second AFLW wooden spoon after finishing last in season six.
Swanson was outstanding early in her 50th match and finished with 24 disposals, four clearances and 394 metres gained to inspire the Eagles to a much-needed victory.
Both teams found goals hard to come by but Eagles midfielder Aisling McCarthy stood out with two majors to help set up the win.
McCarthy also collected 22 disposals to be among the Eagles' best along with Charlotte Thomas (24), Belinda Smith (21) and Isabella Lewis (17, one goal).
"We knew they were going to come out and the momentum was going to swing, and the girls just held on, that was an awesome fight," Swanson told FoxFooty after the match.
"Credit to Port Adelaide as well, they came out with some heat, but I'm super proud of our girls."
The Power (1-4) had their chances to open up a lead in the first half but paid the price for wayward shooting as they went into the main break with only 1.7.
Young gun Abbey Dowrick impressed in an enthralling battle with Swanson in the midfield while making her own impact with 22 disposals and nine clearances.
There was another intriguing matchup between two emerging rucks as the Power's Matilda Scholz used her added height and experience to arguably edge the battle with the Eagles' Lauren Wakfer.
Power superstar Erin Phillips was well held for much of the contest but popped up with a goal on the cusp of the last change that ensured the teams would fight out a tense final term.
West Coast will look to make it two consecutive wins when they face GWS at Blacktown International Sportspark on Saturday, while Port Adelaide host Sydney at Alberton Oval later the same day.
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