Kike de Lucas had given Celta the lead against the run of play four minutes into the second half but a fine jinking run from Antoine Griezmann set up Agirretxe to bring the Basques level just five minutes later.
The comeback was complete on the hour mark as Carlos Vela crossed for Agirretxe to prod home.
The forward had only been on the pitch for nine minutes before he cancelled out Tomer Hemed's 67th-minute strike.
In a bad-tempered encounter which saw eight players booked, Alvaro Vazquez lit the touchpaper by firing the hosts ahead with a screamer just before half-time.
Juan Albin received his marching orders for a second booking after the interval and Antonio Apono levelled the tie from the penalty spot.
A constant menace in the second half, Postiga chipped away at Espanyol's resolve and eventually got his reward two minutes from time to earn Real their first points of the season.
The visitors went ahead within two minutes as Leo's cross was tapped home by Piti, but Betis responded immediately and Jorge Molina headed in his third goal of the campaign moments later.
However, despite withstanding an early Betis onslaught at the beginning of the second period, Rayo looked the more coordinated team throughout and deservedly got the winner on the hour mark when Lass Bangoura teed up Leo to rifle low past Fabricio in the Betis goal.