Javier Hernandez scored twice for Leverkusen but two Daniele De Rossi strikes before half-time made it 2-2.
Miralem Pjanic placed a low free-kick low to give the Italians the lead, before Iago Falque converted for 4-2.
But Kevin Kampl curled in from range and with four minutes to spare, Admir Mehmedi tapped in a dramatic leveller.

Miralem Pjanic
Miralem Pjanic moved Roma 3-2 ahead on 54 minutes
The German side could have won it when Hernandez fired narrowly wide at the end. They sit second in the group on four points, three adrift of leaders Barcelona, who beat BATE Borisov 2-0.
Their coach Roger Schmidt said "this was far from a normal football game", while sporting director Rudi Voller admitted his side "lost the plot" after Roma had scored their first.
Roma - who are without an away win in the Champions League since 2010 - are now bottom of the group and will realistically need to beat Leverkusen in the Italian capital on 4 November to stand a chance of reaching the last-16.