Pienaar celebrated his Goodison Park return by firing the Toffees into a fifth-minute lead on another dismal Premier League afternoon for Chelsea and Stracqualursi doubled the advantage in the second half.
The defeat left Chelsea, losing at the ground for a third successive year, without a win in four league games and was another setback in their bid to retain a top-four place.
Fernando Torres' dismal season also continued as the Spaniard again struggled in front of goal and there was further bad news as England left-back Ashley Cole limped off.
For Everton, who beat Manchester City in their previous home game, the recent revival continued, however.
New signing Nikica Jelavic was unable to feature due to a stomach muscle injury but Pienaar, back at the club on loan after a year at Tottenham, relished being back on Merseyside.
Stracqualursi also proved a handful for the visiting defence and marked a hard-working performance with a first Premier League goal.
Chelsea had welcomed back Cole from suspension and Frank Lampard after injury - relegating new signing Gary Cahill back to the bench - but inspiration was lacking.
Pienaar, by contrast, clearly wanted to make an impact from the outset and burst forward in the first minute in a move that ended with Landon Donovan prodding harmlessly wide.
Chelsea failed to heed that warning and were punished after five minutes when Lampard deflected a Tim Cahill ball over Branislav Ivanovic's head.
Pienaar raced into the box, controlled the bouncing ball on his chest and smashed past Petr Cech into the roof of the net.
Everton quickly looked to build on their lead and David Luiz needed to head behind an awkward Phil Neville cross under pressure from Stracqualursi.
Chelsea's first serious attack petered out as Torres, still without a league goal since September, was forced to turn away from goal and play backwards.
Everton threatened again when Donovan shot powerfully but straight at Cech from distance.
Chelsea were inches away from an equaliser as Daniel Sturridge met a low ball from Juan Mata, which Torres cleverly allowed to run across the edge of the area, with a firm shot but it was deflected over by Neville.
Mata caused further problems for the home defence as he pulled the ball back for Lampard but the England midfielder dragged his shot wide from 18 yards.
Leighton Baines combined well with Pienaar to break into the Chelsea box but Cahill, ruled onside, was unable to control his pass as he turned to shoot.
Baines troubled Chelsea again with an inswinging free-kick but Cahill was flagged offside as he stretched to reach a ball which rebounded off Cech's legs.
Chelsea's Raul Meireles was booked for kicking Stracqualursi as the Argentinian tried to break clear and the Everton striker shot at Cech moments later.
Torres felt he should have had a penalty just before half-time when he was blocked off by John Heitinga.
The former Liverpool forward complained to referee Mike Jones after the whistle was blown but was led away to the tunnel by Everton's Marouane Fellaini.
Chelsea pushed forward in the opening minutes of the second half but Everton's defence, well led by Sylvain Distin, remained firm.
After soaking up the pressure, Everton's Stracqualursi showed great persistence to win the ball from Ivanovic in the Chelsea area but his shot was blocked.
A loud cheer rang out when Torres was booked for a late challenge on Pienaar, whose footwork continued to cause Chelsea problems.
Torres, who had worked hard but without looking like scoring, found himself in the right position when Bosingwa crossed but headed straight at Tim Howard.
Ivanovic was unhappy when Donovan pushed him as the pair ran out of play and his angry reaction earned a rebuke from Jones.
Villas-Boas attempted to inject fresh life into his side by sending on Florent Malouda for Michael Essien, but the change did not appear to be appreciated by all visiting fans.
Everton doubled their lead 18 minutes from time after Neville won the ball in midfield from Cole and released Donovan.
The American ran at the defence and then flicked a fine ball into the path of Stracqualursi, who calmly wrong-footed Cech and slotted the ball home despite the keeper's desperate attempt to claw it away.
Cole had remained on the ground after the challenge by Neville and, after the restart was delayed while he received treatment, he was unable to continue much longer.
Cole looked in much discomfort as he hobbled off to be replaced by Ryan Bertrand.
Chelsea controlled possession in the closing minutes but the Everton defence, with Distin again prominent, coped with everything.