The visitors took an early lead through Gareth Bale, who lashed a first-time shot into the back of the net after Karim Benzema played the ball back to him at the top of the area following a Madrid counter-attack.
Real continued to have the better of proceedings in the first half and Bale nearly doubled the lead for Carlo Ancelotti's side from a long-range free kick that sailed just wide of the goal before the break.
Levante doubled their lead soon after when Jose Campana's half volley from a nice cross on the right beat Courtois for a 2-1 lead.
Real coach Ancelotti was furious with his side's drop in intensity after the break.
"We had everything under control but we're very soft in the second half, that's not what you expect from Real Madrid. We threw away two points today," said the Italian.
"It's difficult to explain, I told the players that any small detail can cost you the game and that's what happened. We leave here with a bad taste because we should have got more than a point after our brilliant first-half performance."
Substitute Vinicius Jr. put Real back on level terms after 70 minutes, outracing the Levante defence to a Casemiro through ball and calmly slotting past Levante keeper Aitor Fernandez to set up a tense finish to the match.
However, Levante retook the lead soon after when Rober netted their third from a set piece with the hosts seeing a potential fourth goal glance off the far post without finding the back of the net.
The see-saw battle continued when Vinicius netted from a neat chip to make it 3-3 right before Aitor was shown straight red for a rash challenge on an onrushing Vinicius, who would have scored if not for the keeper's intervention.
But Vinicius again came to Real's rescue, levelling in the 85th minute with an audacious shot in off the post from a tight angle.
The Brazilian also netted in last week's 4-1 win at Alaves and is joint-top scorer with Atletico Madrid's Angel Correa on three goals from two games after adding some ice-cool finishing to his already impressive skill set.
"I haven't just been work on finishing, I've been working on every aspect of my game to put in a performance like that," said Vinicius.
"We are working a lot on escaping from our markers and Casemiro gave me a great pass for the first goal. For the second I struck the ball with my toe.
"I'm very happy to have started the season well and I've been working hard since day one and I'm not going to change."