Trailing by eight points in the second quarter, Sydney slammed through nine straight goals on their way to a 26.11 (167) to 10.10 (70) win at the SCG on Saturday.
Curnow boosted his league-leading goal tally for the campaign to 62 and was one of 12 Swans goalkickers.
Tom Papley, Jake Lloyd, Logan McDonald and Chad Warner all booted three majors as Sydney registered their biggest score and win over Port.
The 100th AFL game of the Power's Jason Horne-Francis was marred by injury, with the star midfielder hobbling off in the last quarter after his left knee buckled awkwardly when trying to change direction.
The massive win took Sydney six points clear of third-placed Hawthorn, meaning they need to win just one of their last two games to secure second.
They retain a small chance of winning the minor premiership, but need to win their last two matches by big margins and hope ladder leaders Fremantle lose both of their fixtures.
Errol Gulden and Issac Heeney were in the forefront of the Swans' surge.
In the second quarter, Gulden racked up 13 disposals three clearances, three goal assists and six score involvements.
He finished with 39 touches, while Heeney had 24 disposals, seven clearances and three goal assists.
Port, who slumped to a fifth consecutive defeat, started well by winning inside 50s, contested possessions and clearances in the first quarter, which ended with them holding a two-point lead.
Their stocks rose even higher when Mitch Zadow increased their lead to eight by converting a set shot after Tom McCartin hit him high.
That was a good as it got for the visitors, as Sydney lifted and started completing their trademark fast chains with slicker kicks and handballs.
The lead blew out to 50 before the Power kicked the last goal of the quarter and trailed by 46 at halftime.
The Swans booted four goals to one to enjoy a 59-point advantage at three-quarter time and ten to four in the last.
Willem Drew toiled hard for Port and tallied 31 touches and nine clearances, Jase Burgoyne had 34 disposals and seven clearances and Jack Whitlock kicked three goals.
