Playing in front of 37,881 spectators, a record crowd for a quarter-final in Britain, Edinburgh had an early try by Mike Blair and a string of successful kicks from Greig Laidlaw to thank for the historic result.
The home side had to come from behind to overhaul the visitors, who scored an outstanding try through Timoci Matanavou and added nine points from Lionel Beauxis.
Edinburgh made a perfect start and snatched the lead inside two minutes.
Laidlaw launched an up and under from the edge of the Toulouse 22, and when the Frenchmen failed to gather, scrum-half Blair pounced and rolled over to touch down. Laidlaw added the conversion.
Beauxis booted a long-range penalty which just crept over the bar in the fourth minute, and reduced the deficit further with a second 15 minutes later.
Edinburgh lost the services of Allan Jacobsen for 10 minutes when the prop was despatched to the sin-bin for blocking Matanavou as the Toulouse winger attempted to take a quick penalty. However, Beauxis fired the resulting penalty wide.
Within a minute the visitors were ahead for the first time and Edinburgh lost another man to the sin-bin. This time Ross Rennie was the offender and Beauxis made no mistake from close range.
And it got worse for the Scots when Matanavou gathered a speculative kick ahead inside his own 22 and turned on the afterburners to blast his way over for a spectacular solo try.
Jacobsen's return to the fray sparked fresh impetus in the home ranks and when Nick De Luca carried the ball into contact, Laidlaw positioned himself perfectly to take the pass and send a sweetly struck drop goal between the sticks to leave Toulouse 14-10 ahead at the interval.
Blair was forced off with a shoulder injury but Edinburgh made a rousing start to the second period.
Toulouse hooker William Servat committed a blatant offside offence and became the third player to be yellow carded, presenting Laidlaw with a straightforward penalty.
He made no mistake and his next successful kick on 50 minutes fired the hosts into a 16-14 lead.
The four-time champions had an opportunity to strike back instantly, but Beauxis suffered his third miss of the afternoon.
Edinburgh now had their tails up with Tim Visser offering a glimpse of his pace and further pressure yielding a penalty just inside the Toulouse half which proved to be beyond Laidlaw's range.
Toulouse coach Guy Noves rang the changes as the game entered the final quarter, but Edinburgh continued to play the game in the opposition's half.
Nevertheless, the tenuous nature of the two-point advantage meant it was an uncomfortable finale for Edinburgh coach Michael Bradley before Laidlaw slotted a last-minute penalty to spark delirious scenes on and off the pitch.