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.
Tigers coach Damien Hardwick said it was easily the best win of the season.
''It was a terrific effort. It's not often you come from five goals down against a quality outfit such as Sydney," he said.
"They are a side who, normally when you are down, you finish the game at that margin. So for our leaders especially to stand up in that last quarter, it was terrific.
"When you are down they are a very hard side to get back on top of, so for our guys to show some fight and to come out on top was a terrific win and obviously our best win to date."
The leader in the race for the Coleman Medal, Jack Riewoldt, was instrumental all day. He kicked another five goals, made contests and kicked 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 term.
A downside for the Tigers were the turnovers they committed - halfway through the third quarter, the Swans made led 15-3 in scores off turnovers.
Jude Bolton has 32 touches for the Swans, including his usual share of contested possessions, while youngster Daniel Hannebery continued his stellar debut season. The ever-reliable Ryan O'Keefe and Kieren Jack worked hard through the middle, while Rhyce Shaw gave his usual run out of the backline.
Swans boss Paul Roos wasn't interested in talking about September.
"At the moment we're a bottom team so I don't think you can worry about the eight," he said after the match.
"You are what you are - we've got North next week so that's our only focus, and it has to be because we struggled today in the latter half of the game when it was there to be won and we couldn't win it.
"So there's no point in talking about the eight and those sorts of things."
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
The views in this story are those of the author and not necessarily those of the clubs or the AFL.
