South African midfielder Pienaar, who returned to the Toffees late last month from Tottenham Hotspur, opened the scoring with just five minutes on the clock when he sent an effort beyond the reach of Petr Cech from 12 yards out.
Visiting Chelsea offered little in the way of a response before the break, only creating one real chance that saw Everton skipper Phil Neville block Fernando Torres's low drive.
Both sides found goalscoring chances hard to come by after the restart, with neither goalkeeper called into any telling action.
However, with 20 minutes remaining hosting Everton doubled their advantage. American loanee Landon Donovan sent over a cross, which Stracqualursi was able to convert past Cech from close range. The goal was the Argentinian's first in the Premier League.
Andre Villas-Boas's Chelsea struggled to find a way past the Everton rearguard as the pressure continues to mount on the Portuguese boss.