The Austrian finished off a counter-attack in the 94th minute of play at the Estadio de Vallecas to deal a crushing blow to the hosts, who had fought their way back into the game despite playing with 10 men for the final half an hour after Lass Bangoura was sent off.
Ruben Garcia had given Levante the lead against the run of play in the first half, but Argentinian striker Joaquin Larrivey headed in the equaliser after 79 minutes to open his account for Rayo.
Paco Jemez's side went out all guns blazing in search of a late winner, but were dealt a sucker punch by substitute Ivanschitz right at the death.
Iago Falque was handed a start just days after completing his loan move from Tottenham, but his debut lasted just five minutes, calling to be substituted after pulling a muscle.
A goal in the fifth minute of injury time by Richmond Boakye prevented Almeria from claiming their first win of the season as they shared a 2-2 draw with fellow Primera Division newcomers Elche.
Almeria took an early lead through Fernando Soriano only for Edu Albacar to equalise from the spot when Sebastian Dubarbier handled.
The hosts regained the lead on the stroke of half-time when Jose Antonio Verza converted a spot-kick following Cristian Sapunaru's handball.
Almeria appeared to be heading for victory, despite Dubarbier's second-half dismissal after receiving his second yellow card, before Boakye struck to leave both sides searching for their first win.