Hernan Rengifo finished off Luis Tejada's reverse pass in the 67th minute to clinch a 1-1 draw in Peru and despite having Rodrigo Cuba sent off in second-half stoppage time, Juan Aurich held on to claim a point.
The result left River - the 2014 Copa Sudamericana champions - with just two points at the bottom of Group Six.
Juan Aurich (four) leapfrogged San Jose (three) to sit second in the standings behind Tigres UANL (seven).
River arrived in Chiclayo with just a draw and loss from their opening two matches of the 2015 Copa Libertadores, and the visitors looked in good shape when Eder Alvarez headed past Pedro Gallese in the 22nd minute.
But despite taking 24 shots for the match, including eight on target, compared to Juan Aurich's 11, River were unable to wrap up victory.
In Paraguay, Libertad claimed a 1-0 win at home against Estudiantes to move top in Group Seven.
Libertad striker Santiago Trellez ran onto Osmar Molinas' chipped pass just after the hour-mark and stabbed a volley with the outside of his right boot inside the far post.
The Paraguayan club moved to seven points, two ahead of second-placed Atletico Nacional with Estudiantes (four) in third.
'David Beckham was desperate to play in my squad, but I didn't want to bring him along and waste the FA’s money: I had to pick the strongest players': Great Britain coach reflects on Becks snub as 'one of the toughest decisions' of his career
Manchester City report: Bruno Guimaraes and Pep Guardiola discussions revealed, as rumours swirl he could replace Rodri
