Ivi gave the visitors a shock lead after 12 minutes before Lucas Vazquez levelled the scores nine minutes before half-time.
Real failed to take their chances in the second half, and Marcelo received his marching orders late on for kicking out at Jefferson Lerma.
Real started with intent, and Karim Benzema would have had a golden opportunity to open the scoring just five minutes in had he not been denied by a superb tackle from Lerma.
It was the visitors who took the lead, however, when Dani Carvajal allowed Ivi to latch onto a long throw, and the Levante man stroked the ball past Kiko Casilla, per La Liga's official Twitter feed:
Los Blancos looked to reply immediately, but Sergio Postigo blocked Marco Asensio after he'd been picked out by Benzema.
The Frenchman's afternoon was soon over, as he was forced off with an injury inside half an hour and replaced with Gareth Bale.
After Ivi came close to doubling his tally with a free-kick that only narrowly missed the mark, Los Blancos hit back through Vazquez, who tapped home a simple rebound from Sergio Ramos' header following a corner.
Bale almost fired Real in front twice in the dying minutes of the half, heading a Marcelo cross wide from close range before forcing an excellent stop from Raul Fernandez, who was also on hand to keep out Toni Kroos' low effort.
La Liga writer Andrew Gaffney believed the Welshman should have done better:
The winger was again guilty in the second half, heading Vazquez's chipped ball over, while Kroos had another strike saved by Fernandez.
Levante, meanwhile, created little in the way of opportunities aside from an early strike from Alex Alegria, but they remained in the contest as they held firm with resolute defending against Real's attempts to get in front.
Los Blancos nevertheless created a number of late chances, as Raul denied Asensio and Bale missed yet another gilt-edged opportunity—this time a header from Isco's cross. After the keeper made another outstanding save to deny Marcelo in the 89th minute, the Brazilian vented his anger by kicking Lerma while the pair were on the ground.
Kroos rattled the post in injury time, but the visitors hung on for a deserved point.
SB Nation's Lucas Navarrete said Cristiano Ronaldo must take some responsibility for his suspension:
The talisman's absence was of course a huge blow, but Madrid still should have been more than capable of winning.
Real will need to improve significantly when they take on Real Sociedad in their next league outing.