Schalke crushed holders Inter Milan 7-3 on aggregate after Wednesday's 2-1 win in Germany, setting up a clash with Manchester United.
Rangnick, nicknamed the professor, took over from Felix Magath in late March and his team have since flourished, winning all four games under him.
There was never any doubt that Magath's infamous gruelling training regime would see Schalke players still in outstanding physical form towards the end of the season.
It was their mental strength that was a major stumbling block, however, as last year's Bundesliga runners-up spent most of the season struggling to stay clear of the drop zone with Magath fighting running battles with fans, club bosses and his under-performing players.
"What can I say about such a team," 52-year-old Rangnick beamed after their second win over Inter, following a 5-2 demolition in the first leg in Milan last week.
"I am delighted that the team gave itself a reward with a second win and was not just looking to advance. What a performance," he said.
"Everyone ran their socks off for the other players and that was the key to our success."
Central defender Christoph Metzelder, who joined last year from Real Madrid and had yet to peak this season, played arguably his best game on Wednesday, shutting out Inter forwards Diego Milito and Samuel Eto'o.
Under Magath, the former Germany international's confidence level was so low he had to seek psychological help.
Farfan, suspended in the return leg, orchestrated Schalke's offensive explosion in the first leg and now wants to stay on at Schalke.
"(Under Rangnick) we have become a real team with everyone fighting for the other," said defender Benedikt Howedes, who fired in the winner on Wednesday after Spaniard Raul set him up with a lob over a static defence. Raul had also scored Schalke's first goal.
"We have an amazing determination on the pitch and we just radiate danger up front," he said.
