Santa Cruz, on loan from English giants City, scored the first of a double after seven minutes and Salva Sevilla extended the lead from the penalty spot before Efrain Juarez pulled a goal back for Zaragoza.
Betis, promoted from the second division last season, never looked secure at the back but another penalty this time from Benat Etxebarria and Santa Cruz's second on 49 minutes gave them a comfortable cushion.
However, midway through the second half, Betis keeper Casto Espinosa rashly darted off his line and was red-carded for upending Luis Garcia despite having defenders behind him.
A double from Juan Carlos led to some anxious last minutes for Betis but they held on.
Super-sub Walter Pandiani scored an injury time winner as Espanyol beat Getafe 1-0 and condemned their rivals to their third successive defeat.
Getafe had not won away since January and their confidence was not high after a poor start to the season but they were up against an Espanyol team who had also suffered two defeats from three games.
There was plenty of toil but little flowing football in the first half.
However, after the break Espanyol took the game to the visitors.
Javi Marquez saw a deflected shot go just wide and Alberto Lopo blocked a Vladimir Weiss strike on the goal line but it was really keeper Miguel Moya who was keeping Espanyol at bay.
In particular he did well to stop a Thievy Bifouma shot from close range.
Getafe missed a glaring chance themselves when Juan Valera blazed wide late on but Espanyol got a deserved win a minute into injury time, when Pandiani coolly slotted home in the box.
Spain's top two continued to not have it their own way in midweek as Barcelona could only draw 2-2 at Valencia while Real Madrid came away with a 0-0 draw from their trip to Racing Santander.
Cesc Fabregas rescued a point for Barcelona and has now scored in the four league games so far to set a record for a new player, but the result means the champions have now already dropped four points from the opening fixtures.
It has been an even worse start for Real whose draw followed a defeat away to Levante at the weekend.
Coach Jose Mourinho rang the changes which included Jose Callejon and Lassana Diarra coming into the side but it was another ineffective performance.
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