The hosts opened up a two-goal lead in the first 19 minutes of the hugely entertaining encounter, through Mexico forward Javier Hernandez.
Roma raced back with two goals from Daniele De Rossi and doubled the tally in the second half before another twist in the game produced two late strikes for Leverkusen, with Kevin Kampl scoring in the 84th and Admir Mehmedi 112 seconds later, to rescue a point.
"We could have won 5-4 in the end but we cannot be satisfied with how we played for long periods of the game," said Leverkusen keeper Bernd Leno, who made a string of good saves. "I can't really explain what happened in the end."
Leverkusen remain second on four points with Roma, still without a win, bottom on two. Barcelona are top on seven after beating BATE Borisov 2-0.
Leverkusen struck after just four minutes with Hernandez sending keeper Wojciech Szczesny the wrong way to score with a penalty, awarded for a handball by Vassilis Torosidis.
The Italians, with their backs to the wall after only one point from their opening two matches, initially struggled with the high-pressing Leverkusen and Hernandez, who joined from Manchester United this season, snatched his second goal from a pinpoint Hakan Calhanoglu pass in the 19th.
Roma's defence looked as shaky as in their 3-2 loss to BATE in their previous game, but De Rossi cut the deficit in the 29th, tipping the game instantly in Roma's favour.
The 32-year-old, who also scored on Saturday to mark his 500th appearance for the club in their 3-1 win over Empoli, then slipped into the box to connect with a Miralem Pjanic free kick to equalise in the 38th with Leverkusen having inexplicably surrendered control of the game to the Italians.
Roma picked up where they left off and a scintillating run by Gervinho required another good save from Leno but the keeper was helpless when Pjanic curled his sublime free kick over the wall.
Torosidis should have added another in the 70th but Spaniard Falque made amends in the 73rd, tapping in from a Gervinho assist as Roma looked to stage a memorable comeback. But their celebrations came too early with Leverkusen staging a last-gasp rally for an unusual high-score draw.
"We are disappointed but the result remains positive under the circumstances," Roma coach Rudi Garcia told reporters. "We played well after the double blow and even managed to score four goals."
"It was a great game for those who were at home watching this on television, but not for the two coaches," he said.
(Reporting by Karolos Grohmann; editing by Clare Lovell)
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