Inter failed to build on a first-minute lead and lost their shape completely as Schalke ran riot before the home side finished with 10 men after defender Cristian Chivu was sent off.
GEAR:Free printing andup to £20 off Champions League kits. Free delivery on orders over £50
But the Germans struck back through Joel Matip and, although Diego Milito briefly restored Inter's lead, two goals from Edu, one from Raul and an own goal by Andrea Ranocchia put Schalke in complete control of the tie before next week's second leg.
"It could have been 10-5 or 8-4," said Schalke 04 coach Ralf Rangnick. "Our attacking policy paid off. We looked dangerous with every attack, we always looked as if we were going to get a shot on goal."
Inter coach Leonardo, already smarting after a 3-0 defeat by rivals AC Milan in Serie A on Saturday, added: "We have to change this situation, we lost two important games, we must stay together and we must try to forget these two heavy defeats.
"I have to be realistic but we are going to do our best and try and get the result we need (in the second leg) even if it's a very difficult one.
"Anything is possible in football, we will do our best to the results."
The match had an extraordinary start when Neuer rushed out of his area to head away Esteban Cambiasso's long ball forward.
The visitors levelled in the 17th minute when Kyriakos Papadopoulos's effort was saved by Julio Cesar but Matip followed up to bury the rebound.
Inter made an electric start to the second half, Milito failing to put away a clear chance after he scuttled clear of the Schalke defence and Neuer atoning for his earlier eccentricity with a superb save to deny Samuel Eto'o.
