First-half goals from Robert Rojas and Rafael Santos Borre had River on track to sensationally overturn a 3-0 deficit in the semi-finals in Sao Paulo.
Champion in 2018, River had a goal ruled out early in the second half and a penalty call overturned after Rojas' red card, Palmeiras holding on for a 3-2 aggregate victory.
River found its way back into the tie before half-time.
Rojas produced a brilliant header from a Nicolas De La Cruz corner before Borre nodded in a Matias Suarez cross in the 44th minute.
River thought it had levelled the tie in the 52nd minute, but Gonzalo Montiel's volley was ruled out for offside after a VAR check.
There was more drama to follow.
Rojas was sent off in the 73rd minute after being shown a second yellow card.
But River was awarded a penalty just two minutes later for a challenge by Alan Empereur on Suarez, only for the decision to be overturned after a VAR check.
There was another check for a penalty for River in the 100th minute, but Palmeiras held on to move into the final against either Santos or Boca Juniors.