Following early protests over late kick-off times on Friday, Levante took an early lead at the Estadio Teresa Rivero when Ruben Garcia displayed brilliant close control before sliding the ball home on 26 minutes.
Rayo's attempts to get back into the game were dealt a blow when Lass Bangoura was given a straight red for kicking out at Juanfran on the hour mark.
However, despite the man disadvantage, Rayo equalised 12 minutes from time with Joaquin Larrivey's header from Roberto Trashorras's deep cross squaring the ledger for the home side.
Having levelled the match at 1-1, Rayo retreated in an attempt to save a point but in the fourth minute of stoppage time, Jordi Xumetra orchestrated one last attack for Levante, finding Ivanschitz to score a last-gasp winner and breathe life into Levante's campaign.
Elsewhere, La Liga newboys Elche netted a late equaliser through Richmond Boakye, just days after his loan signing from Juventus was completed.
Elche twice fought back from behind to draw 2-2 with Almeria, who led through Fernando Soriano and again on the stroke of half-time, when Verza converted from the penalty spot.
Edu Albacar netted Elche's goal in the first half, also from the penalty spot, and the visitors collected a point in Almeria through Boakye's 95th-minute equaliser after the hosts' Sebastian Dubarbier was sent off with 23 minutes left.
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