European champions Toulouse stuttered to an unconvincing 20-13 victory over Edinburgh in their pool match.
The Scots, meanwhile, must now beat Toulouse on Friday and seal victory over Wasps to have any chance of reaching the knock-out stages.
They had been trailing by just one point going into the final 10 minutes before late points from the boot of Frederic Michalak.
The France international slotted over a late penalty and drop goal for the win.
After some early sustained pressure, the Gunners looked to have opened the scoring courtesy of an Ally Hogg try.
But the seemingly-legitimate score was disallowed after referee Dave Pearson decided he had lost control of the ball.
Moments later, Michalak slotted over a penalty before combining with Yannick Jauzion and Jean-Baptiste Roux to send Vincent Clerc in to score.
Michalak missed the conversion but added another two penalties to give Toulouse a 14-point lead.
But Edinburgh got back into contention when a sniping Mike Blair touched down after breaking from the base of his pack.
Chris Paterson added the conversion then two more spot kicks to set up an exciting finale.
But he was eventually upstaged by Michalak's two late kicks.
"It was definitely a game we could have won, but we didn't get the breaks at crucial moments.
"After a poor opening spell, we had enough territory and possession to earn a positive outcome, but it wasn't to be.
"One or two refereeing verdicts went against us, and we should have used the ball more wisely.
"The bottom line, however, is that we are able to compete with the bigger teams in the tournament and we will keep building."
Edinburgh: Southwell, Paterson, Di Rollo, Jorgensen, Webster, Godman, Blair, Hewett, Hall, Smith, Strokosch, Murray, Mustchin, Hogg, Taylor. Replacements: Kelly, Jacobsen, Cross, MacDonald, Lawson, Pyke, Dey.
Toulouse: Thomas, Clerc, Jauzion, Fritz, Heymans, Michalak, Elissalde, Poux, Bru, Hassanein, Brennan, Millochlusky, Bouihlou, Nyanga, F. Maka. Replacements: Lacombe, Menkarska, Ledevedec, I. Maka, Dubois, Montauriol, Poitrenaud.