Joao Moutinho broke the deadlock in the 61st minute from former Leverkusen striker Dimitar Berbatov's assist as the Bundesliga leaders were made to pay for their profligacy.
A first-half goal by AS Monaco striker James Rodriguez secured all three points for the Colombians, who are now guaranteed at least a play-off spot from South America's nine-team qualifying competition.
The 33-year-old, who left Spanish La Liga champions Barcelona at the end of last season following six years in Catalonia, has an option for a second season.
In Monaco, former Atletico Madrid striker Falcao will team up with another Colombian, winger James Rodriguez, who has agreed to move to the Principality from Porto.
Porto will make a massive profit on both, with Moutinho costing 11 million euros when he signed from Sporting in July 2010. Rodriguez signed the same month from Argentine club Banfield for 5.1 million euros.
James Rodriguez's 90th-minute goal gave FC Porto a 2-0 win at third-place Braga to extend its unbeaten streak to 23 straight matches in all competitions and retain its share of the Portuguese league lead.
