The 'Flying Dutchman', who will qualify to play for Scotland at the end of the season, scored a try after 35 seconds and another four minutes from time as Edinburgh sensationally came from behind to triumph and claim a second win from two games following last weekend's win at London Irish.
The hosts scored 17 points in the opening 10 minutes but conceded 28 unanswered points and were trailing by 20 with 14 minutes remaining.
But Racing replacement Jone Qovu was sin-binned with 11 minutes to go and Edinburgh scored two converted tries to triumph, with captain Greig Laidlaw kicking the winning conversion.
Edinburgh's tries came from Visser (two), Netani Talei, Tom Brown, Roddy Grant and Laidlaw, who added six conversions and two penalties in a 23-point haul.
Racing finished with five tries - for Julien Saubade, Juan Imhoff, Henry Chavancy (two) and Jonathan Wisniewski, who kicked 20 points, while Juan Martin Hernandez kicked one conversion but missed a drop-goal attempt with the last play of the game.
It was not a night to be a defence coach, with missed tackles commonplace.
Edinburgh kicked off, but Racing returned the ball high allowing the hosts to spread the ball wide where Visser did the rest. He chipped over Saubade and won the dash for the line to score his sixth try of the season.
Wisniewski and Laidlaw then exchanged penalties before Edinburgh crossed for a second try.
Matt Scott and Geoff Cross forged forward before Visser opted to chip again. He was taken out by Imhoff, but Laidlaw scooped up the loose ball to score and converted himself.
The frenetic start continued when Racing scored two converted tries.
Saubade went over on the right following an overlap and Imhoff touched down following his team-mate's inside pass and the score was allowed despite a suspicion of a forward pass in the build-up.
Edinburgh went behind for the first time as Wisniewski waltzed over on the left and converted himself after 22 minutes.
Racing exposed Edinburgh's poor defence out wide again as Chavancy burst through a gap and Laidlaw missed a covering tackle, allowing the centre to go under the posts for a converted score.
A Laidlaw penalty on the stroke of half-time saw Edinburgh cut the scoreline to 31-20 at the break.
Scrum-half Mike Blair replaced Harry Leonard at the break, with Laidlaw switching to fly-half.
Wisniewski kicked a penalty and Racing scored their fifth try after 47 minutes when Chavancy gathered Saubade's kick ahead and touched down unopposed after James King, on for Scott, lost the ball in a tackle in midfield.
Wisniewski converted and added a penalty before Edinburgh responded.
Edinburgh punched forward down the middle with a strong Allan Jacobsen run, before spreading the ball wide and Stuart McInally gave the scoring pass for back-row colleague Talei to touch down.
From the restart Edinburgh infringed and Wisniewski kicked his fourth penalty before Laidlaw sent Brown over with Edinburgh's next attack.
The half-back's conversion cut the deficit and Racing were reduced to 14 men after 69 minutes when Qovu was sin-binned for striking.
Seven minutes from time Grant went over for Edinburgh's fifth try after good runs by Brown and King and Laidlaw cut the deficit to six points.
And Edinburgh used their extra man well, spreading Racing the width of the pitch time and again before Visser touched down on the left for his second try.
Laidlaw converted for a one-point lead, but Edinburgh had to survive the remaining three minutes of play.
And Hernandez fluffed a drop-goal attempt with the final play of the game as the hosts emerged triumphant.
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