Paterson kicked two conversions and a penalty but had to be replaced in the second half, because of a foot injury.
Having established a 12-10 half-time lead, Leinster crumbled in the second half as poor defending allowed Visser, Robertson and man-of-the-match Blair all touch down in the space of 15 minutes.
Blair added a penalty to put 20 points between the sides, yet Leinster rallied to get within touching distance of a losing bonus point.
Isa Nacewa converted a try from replacement Jamie Heaslip, and while Rob Kearney got over in the dying seconds, Ian Madigan, fresh off the visitors' bench, failed to land the difficult conversion.
The concession of two late tries could not take the gloss off what was a much-needed victory for Edinburgh, their first in eight Magners League matches.
The Scots had Argentinian lock Esteban Lozada and flanker Ross Rennie back in harness, with team captain Roddy Grant moving to number 8 in place of dead leg victim Netani Talei.
Leinster coach Joe Schmidt welcomed back Shane Horgan and Brian O'Driscoll into the back-line, while South African duo Heinke van der Merwe and Richardt Strauss, Devin Toner, Dominic Ryan and Stephen Keogh all returned to the starting pack.
The Irish province had to cope with the late withdrawal of Jonathan Sexton, which saw Nacewa don the number 10 jersey and youngster Madigan promoted to the replacements bench.
Paterson and Nacewa traded early penalties as the sides scrapped for an early foothold, but defences were on top for the most part.
A quality 14th minute strike from Nacewa, from 40-plus metres, sent Leinster in front for the first time, and they pushed 9-3 clear when full-back Kearney crowned his 100th provincial cap by firing a monster penalty kick over from inside his own half.
However, Edinburgh got on top in the second quarter as they began to carry a try-scoring threat. Paterson sent a penalty to touch and when Lozada won the resulting lineout, the hosts kept it tight before finding a gap out on the right.
A sweet pass back inside from Ross Ford put his fellow forward Scott Macleod haring towards the corner. He was held up by a try-saving tackle from Luke Fitzgerald. Edinburgh kept the pressure on though, and it yielded a try under the posts for prop Cross who took a backhanded pass from Mike Blair and barged past Toner to score.
Paterson's successful conversion was cancelled out by another long range penalty from Nacewa, whose 28th minute effort nicked the crossbar before going over.
Leinster were creating very little in attack at this point, and Edinburgh, with Ford, Macleod and Grant working their socks off, were proving the more aggressive in contact situations.
Rob Moffat's charges pinned Leinster back for a period, but a promising move, which Lozada and Allan Jacobsen added their weight to, was ended by the concession of a penalty close to the Leinster posts.
Hooker Strauss was one of the busiest Leinster forwards, his accurate lineout throwing helping to relieve some pressure as Edinburgh looked to strike.
They almost engineered a try for danger man Visser. Scrum half Blair and the energetic Cross added great urgency to an attack and as the Scots probed the left wing, Leinster flanker Ryan did brilliantly to prevent Visser from scoring in the corner.
Edinburgh did have a late chance to retake the lead, but Paterson was off target after Leinster's stand-in captain Shane Jennings was pinged for not rolling away after the tackle.
The Scotland international uncharacteristically missed a second penalty in the early stages of the second half, before Edinburgh put on their try-scoring blitz.
A powerful burst down the right from Ben Cairns created the first incision in the Leinster defence, Cross rumbled up close to the try-line and a quick switch of play to the left ended with Visser pivoting past Horgan for his fifth try of the campaign.
Paterson missed the conversion but it was his surging run into the visitors' 22 which laid the platform for Edinburgh's third try. Leinster's defence was all at sea as Robertson, in acres of space, was sent through by David Blair's looping pass.
Paterson converted to pass the 700-point mark and give Edinburgh a 22-12 lead that their spirited efforts deserved.
The increasingly influential David Blair missed a drop goal, but Leinster's long range counters were amounting to little and Moffat's side chalked up their bonus point with relative ease.
On the hour mark, Heaslip, a much-needed presence in the Leinster pack, was penalised for not rolling away. The elder Blair spotted an opening in the 22, took a quick tap and released his younger brother David who made it over in the corner past Eoin Reddan.
The try scorer showed his place-kicking class by adding the conversion from close to the touchline, and Edinburgh were almost certainly home and hosed at 29-12.
Leinster improved their general play in the final quarter, helped by the likes of Heaslip, Isaac Boss and young hooker Jason Harris-Wright.
Heaslip was held up close to the Edinburgh whitewash, and luck was not on their side when Ryan subsequently lost the ball in contact and Cian Healy, another replacement, was robbed by Rennie as he tried to set up an attack.
Leinster's scrum gave way in the 68th minute, allowing David Blair tag on another three points for his side.
Edinburgh's concentration waned in the closing minutes, however, as a lovely break and deft offload from Fitzgerald led to Heaslip dotting down. Nacewa converted before being replaced.
Another converted try would have garnered a losing bonus point for fast-finishing Leinster. Replacement Mariano Galarza went desperately close as he wrestled up way towards the Edinburgh posts.
The visitors maintained the momentum and a string of passes out to the left created a try in the corner for Kearney.
That handed Madigan a chance to seize the losing bonus point with the kick. But, from a very difficult position, his attempt was never on target and Edinburgh were left to celebrate a long-awaited win.
After watching his side end their losing run, Edinburgh coach Rob Moffat said: "It was a good victory but there are plenty of aspects that will need to be better.
"What was pleasing was the way we came out at the start of the second half and played the game at a tempo that enabled us to get the try-scoring bonus point."
Summing up the visitors' feelings, Leinster boss Joe Schmidt admitted: "It was a disappointing effort from us tonight and Edinburgh were a bit more desperate (for the win) than us.
"The intention was there from us, but when they got on top they bullied us. We didn't play a lot in our half and because of that we couldn't set a platform. Credit to Edinburgh, they looked after the ball pretty well throughout."
|Edinburgh Rugby Score Card|
|Leinster Rugby Score Card|