Cavani scored his second goal in three qualifiers before Stuani grabbed the second in the space of four minutes to put Uruguay in fifth place in the group, behind Ecuador only on goal difference.
Colombia, who went into the match looking for the one point they need to seal their ticket to the 2014 Brazil finals, remain second and host third-placed Chile next on October 11.
Ecuador were held to a 1-1 draw by eliminated Bolivia in La Paz earlier on Tuesday and leaders Argentina were playing Paraguay later in Asuncion.
The top four teams at the end of the campaign next month go through to the finals in Brazil with the team in fifth place meeting Jordan in a two-legged playoff in November for one more berth.
Uruguay had lost 4-0 to Colombia when the two teams met in Barranquilla exactly a year ago, a defeat that sparked a run in which they took only two out of a possible 18 points and stared elimination in the face.
The 2010 semi-finalists in South Africa have revived their chances by winning their last three qualifiers.
Colombia had the better of the first half and the first good chance fell to Radamel Falcao from a fine through ball by midfielder Fredy Guarin but the striker shot wide past advancing goalkeeper Fernando Muslera.
Muslera was also called into making a diving save from midfielder James Rodriguez.
Uruguay, who had made five changes due to suspensions and injuries, served notice of their intentions with shots by Cavani and Luis Suarez that hit David Ospina’s woodwork either side of the interval.
Coach Oscar Tabarez’s masterful substitutions finally bore fruit in the last quarter of an hour.
Right back Maxi Pereira sent in a high cross and Cavani rose between the central defenders to head past Ospina in the 77th minute.
Less than four minutes later, midfielder Gaston Ramirez chested down a throw in and turned and passed into the middle where fellow substitute Stuani slipped between two defenders to beat Ospina.
Uruguay meet Ecuador in Quito next month in a match that could settle which team meets Jordan before ending their group campaign at home to Argentina.
(Reporting by Martin Rodriguez Writing by Rex Gowar, editing by Nick Mulvenney)