Neymar on Thursday scored the opener at the Estadio Olimpico by converting a 72nd-minute spot kick after Jesus was felled by goalkeeper Alexander Dominguez, reports Xinhua.
Juan Carlos Paredes was sent off shortly after when he received a second yellow card for a challenge on Renato Augusto.
Jesus then scored twice in five minutes, giving former Corinthians coach Tite a dream start in his first match in charge of the Selecao. It was also Brazil's first victory in a World Cup qualifier in Quito.
The result means Brazil now have 12 points from seven qualifiers, one point less than Ecuador, who have lost their previous two matches.
Many predicted Brazil would struggle in the Ecuadorian capital, nestled in the Andes at 2,850m above sea level. Instead, it was Ecuador who appeared uncomfortable in the rarefied air.
Brazil looked more likely to score from the outset with recalled Guangzhou Evergrande midfielder Paulinho involved in many of the Selecao's best attacking moves.
However, it was actually the 61st-minute substitution of Chelsea midfielder Willian for Liverpool's Philippe Coutinho that provided the spark Brazil needed.
Coutinho twice went close to scoring in his first five minutes on the pitch; once with a long-range bullet and another after darting into the area and lashing a shot that Dominguez somehow kept out with his leg.
Jesus then demonstrated why Manchester City paid 27 million pounds ($35.8 million) for his signature last month.
The 19-year-old ghosted past Ecuador's defence before being tripped by Dominguez when through on goal.
With his trademark hunch-backed run-up, Neymar waited for the keeper to commit before stroking his penalty the other way.
Minutes later Jesus sent the crowd into stunned silence, netting his first international goal with a sublime back-heel flick after Marcelo's cross.
The striker's second goal was even more impressive as he expertly controlled a Neymar cross at the top of the box and then, in the same motion, turned and fired an unstoppable shot into the top corner.
Other winners in a fascinating round of South American zone qualifiers on Thursday were Bolivia, Colombia, Argentina and Paraguay.
Lionel Messi, who quit international football in June before retracting his decision last month, handed 10-man Argentina a 1-0 victory over Uruguay in Mendoza.
The Barcelona forward netted with a deflected shot just before half-time at the Malvinas Argentinas stadium to lift Argentina to the top of the 10-team standings.
The Albiceleste had to defend grimly for the entire second half after Juventus forward Paulo Dybala was shown his second yellow card for a rash challenge on the stroke of half-time.
Argentina now have 14 points from seven matches, one point ahead of second-placed Uruguay.
In other fixtures on Thursday, Real Madrid's James Rodriguez was on target in Colombia's 2-0 victory against Venezuela, Bolivia beat Peru 2-0 and Paraguay overcame reigning Copa America champions Chile 2-1.
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