Alvaro Negredo was first to miss from the spot on seven minutes after Jordi Amat had pulled down the Spanish international inside the area, but he struck the post and the ball rebounded to safety.
But the visitors were handed another opportunity 20 minutes from time when Ivan Rakitic was brought down by Ruben Martinez.
The goalkeeper was shown a red card but his replacement Dani Gimenez did not even have to make a save as Rakitic pulled his penalty wide.
And despite referee Clos Gomez further infuriating the home fans by adding on six minutes for stoppages, Rayo held on to move joint top of the table with seven points from their first three games.
Sevilla themselves remain unbeaten but will regret dropping two points having controlled the game from start to finish.
Even after Negredo's early penalty miss, Michel's side should have been ahead within the first 20 minutes as Manu del Moral was denied by a flying save from Ruben before the same player passed up a glorious opportunity after a fine run and cross by Jesus Navas.
Ruben saved Rayo once more just before the break when Tito inadvertently turned Rakitic's free-kick towards his own goal, but barring a Roberto Trashorras free-kick that Andres Palop did well to turn over, Sevilla continued to look the more threatening side in the second period.
And their best chance to win finally came when some slack play by the Rayo defence allowed Rakitic to steal the ball just outside the area and the Croatian was cleaned out by Ruben as he tried to go around the keeper.
A lengthy delay followed as Dani prepared to come on, and that perhaps affected Rakitic as his poor penalty slipped past a post.
Sevilla continued to press for a winner but the closest they came was when Alberto Botia headed over from Rakitic's corner with three minutes remaining.