The other tie played on Wednesday also ended level as exuberant Universidad de Chile, who came from a 4-1 first-leg deficit to thrash Deportivo Quito 6-0 in the previous round, drew 1-1 at Paraguay's Libertad.
Universidad de Chile finished with 10 men after substitute Emilio Hernandez, who came on in the 80th minute, was dismissed only nine minutes later for a bad tackle on Sergio Aquino.
Vasco and Brazilian champions Corinthians barely mustered a scoring chance between them in driving rain and on a muddy pitch in Rio de Janeiro's Sao Januario stadium.
Vasco's Alecsandro had the ball in the net in one of the few openings but his second-half header was controversially disallowed for offside.
The result left Corinthians, who are one of Brazil's biggest clubs but have never won South America's top club competition, as slight favourites for next week's second leg in Sao Paulo.
In the other tie, Victor Caceres gave Libertad, coached by Argentine World Cup winner Jorge Burruchaga, a seventh-minute lead at the Nicolas Leoz stadium in Asuncion.
Chilean champions Universidad de Chile, whose coach Jorge Sampaoli has been heavily influenced by Athletic Bilbao's Marcelo Bielsa, have been widely praised for their fluid attacking game but this time, the match developed into another physical battle.
However, the visitors levelled 10 minutes into the second half with a deflected shot from Gustavo Lorenzetti.
The other ties, both Argentina-Brazil clashes, are on Thursday in Buenos Aires with Boca Juniors hosting Fluminense and Velez Sarsfield entertaining defending champions Santos.