The four-time World Cup champion's hopes of progressing from Group E are still alive thanks to Niclas Fullkrug's goal in Germany's draw with Spain.
Spain, who hit seven past Costa Rica in their opening win, looked dangerous immediately, with Germany keeper Neuer called into action to keep out a thunderbolt by Olmo in the 7th minute.
Every team can make it to the round of 16 in the World Cup.
Germany looked to turn around their World Cup fortunes after a shocking 2-1 loss to Japan to open group play on Thursday, but Serge Gnabry missed a shot in the 25th minute.
Spain continued to look the better of the two teams going forward but lacked an end product, while Germany had a goal ruled out after a VAR check.
Simon was forced into his first save in the 56th minute when the 2010 champion got into trouble trying to play out of the back, allowing Kimmich to shoot from just inside the penalty area, but the Spain keeper made an impressive save.
After being brought off the bench early in the second half, Alvaro Morata was able to beat his marker to Jordi Alba's near-cross and head it past Neuer to give Spain the lead.
Germany, who hadn't gone winless for three straight World Cup games in 60 years, fought hard and should have drawn level in the 73rd minute but Simon made a great save.
With time running out on Hansi Flick's side, Fullkrug had the ball fortuitously fall to him in the Spain penalty area and he blasted a beautiful strike past Simon to give Germany their first point of the tournament so far.
We were well aware of what was at stake. The teams were level but Spain had a chance to win it in the end.
Things come to you when you win. It might be the spark we need.
The result keeps Spain atop the group on four points and Germany in last with just one, but they can advance from a mix of scenarios with the most ideal being a win over Costa Rica and a win by La Roja against Japan.
Luis Enrique said that there was a strange feeling in the changing room because they had the chance to beat Germany.
We have to believe that we are at the top of the group of death.
We will not speculate on the outcome of the Japan game on Thursday. We will go for the win in order to get the top spot.