The win lifts the side off the bottom of the ladder for the first time since round two, and has given the Tigers four wins in their last five games.
The Swans led by 33 points midway through the third, but the Tigers kicked eight of the last 10 goals, with the final one to new star forward Andrew Collins, who kicked three for the day, all in the second half.
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The leader in the race for the Coleman Medal, Jack Riewoldt, was instrumental all day, either kicking goals (five), making contests or kicking it long to advantage inside 50.
Tigers defender Kel Moore was terrific on Adam Goodes, whose form slump continued, while skipper Chris Newman used the ball beautifully all day.
Richmond hearts were in mouths in the second term when Tigers star Brett Deledio went straight to the rooms with what looked to be a badly hyper-extended elbow, but he was soon back on the ground after having the elbow heavily strapped, and was instrumental in the win.
The first 10 goals were scored turnabout, with 10 lead changes, but then the Swans kicked four in a row, and got a handy buffer over the Tigers in the third term.
Both sides put huge pressure on early, but it looked as though the more mature Swans' bodies were the ones who were able to keep working harder for longer.
But the Tigers kicked back into gear, and rolled inexorably towards the finish, with Newman and Collins kicking a couple each in the last.
A downside for the Tigers was that they coughed the ball up far too often, and the Swans made them pay - halfway through the third, the Swans led 15-3 in scores off turnovers.
Richmond 3.0 5.0 9.3 14.5 (89)
Sydney Swans 3.3 6.6 10.11 12.13 (85)
GOALS
Richmond: Riewoldt 5, Collins 3, Newman 2, Farmer, Nason, Griffiths, Martin.
Sydney Swans: McVeigh 3, Moore 2, Goodes 2, Dennis-Lane, Jack, Pyke, McGlynn, Mumford.
BEST
Richmond: Deledio, Moore, Riewoldt, Collins, Newman, Astbury, Connors, Jackson.
Sydney Swans: Bolton, McVeigh, Hannebery, O'Keefe, Jack, Bird, Malceski
INJURIES
Richmond: Deledio (elbow)
Sydney Swans: Playfair (back)
Reports: Nil
Umpires: Rosebury, Stewart, Grun
Official crowd: 39,386 at MCG
