Uruguay's Forlan scored both goals, the second just four minutes from the end of extra-time in a game that crackled with excitement from the start, after being set up on each occasion by his Argentine team-mate.
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Fulham showed all the strength of character that had earned them their first European final appearance and came agonisingly close to a penalty shoot-out that might just have delivered them the title in the first season of the rebranded UEFA Cup.
Instead an epic journey that began back in July and took in 19 matches ended in heartbreaking style.
They had recovered from Forlan's first goal in the 32nd minute, with Simon Davies popping up unmarked at the back post to squeeze in a first-time shot for an equaliser five minutes later, and lived with their more fancied opponents comfortably enough until Forlan's second sealed their fate.
"Unfortunately, Diego Forlan popped up to score a goalscorer's goal in the first half and again at the end, and then there was no time to recover," Fulham manager Roy Hodgson told reporters. "It was a bitter disappointment."
FAMILIAR GROUND
That disappointment was a familiar experience for Hodgson, who also tasted UEFA Cup final defeat 13 years ago when his Inter Milan side lost on penalties against Schalke 04.
For Atletico, it was another glorious night in Germany following their only other European final success, in a replay against Fiorentina in Stuttgart in 1962.
It also delivered a first major title of any sort since their double-winning season of 1996 and they still have a King's Cup final to come.
"I told the players to be courageous and not be lazy," said Atletico coach Quique Sanchez Flores. "I said they should be proud and they should remember how they'd dreamt of winning cups, and to go out and win this one."
Fulham, a modest London club with precious little European experience before this year, took time to settle in front of the 49,000 crowd, despite the confidence that victories over the likes of Juventus and Hamburg in earlier rounds had given them.
Aguero always looked like troubling them, with the Argentine's ability to keep the ball at his toes and wriggle into space in their penalty area a constant worry.
There had been several close calls before he set up the opener, with a little bit of luck it must be said, as he controlled the ball on the edge of the box, seemed to mishit a shot across goal and watched as Forlan snapped it up.
After the equaliser from Davies came almost immediately the game drifted a little, as both teams became more cautious.
Fulham had their best spell at the start of the second half but rarely threatened Atletico's cool 19-year-old keeper David de Gea.
The second half fizzled out as the drizzle returned. Still Aguero looked the man most likely to create something out of nothing and so it proved.
As the game stretched long into extra-time Aguero chased down yet another lost cause, beat his marker for the umpteenth time and sent in a low cross that Forlan diverted in expertly.
Fulham players slumped to the ground and barely managed to rouse themselves for a couple of hopeless punts in the general direction of the Atletico goal.
"Even if we'd had the legs to get back, we didn't have time," said Hodgson. "We had quite a few players running on empty."
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