Squillaci rose high to connect with a deep Adriano cross to give Sevilla the lead against the run of play in the 23rd minute and Jesus Navas made it 2-0 following a poor clearance by goalkeeper Jens Lehmann 10 minutes after the restart.
Squillaci headed his second goal after 72 minutes, having again been left unmarked at the far post, before substitute Elson grabbed a consolation for the hosts two minutes later.
Sevilla's win puts them on nine points after three games and extends their lead at the top to five. Stuttgart slipped to third spot on two points, one ahead of Rangers who were thrashed 4-1 by Unirea Urziceni who are now second on four points.
"Team solidarity won today," Sevilla coach Manolo Jimenez told reporters. "Stuttgart dominated the first half and we knew we had to be patient. The team read the match very well."
Sevilla, third in La Liga after a spectacular start to the season, had grabbed the lead against the run of play, with Stuttgart having largely monopolised possession.
BRIGHT START
The hosts had begun brightly with Aleksandr Hleb thriving in a playmaker role and almost creating a goal after 11 minutes.
The Belarus international fed Cacau on the left whose low drive was blocked by Sevilla goalkeeper Javi Varas, called into the side to replace the injured Andres Palop.
Varas was beaten two minutes later when Stuttgart's Zdravko Kuzmanovic fired into the net but the effort was ruled offside as defender Khalid Boulahrouz was left stranded in the box.
Sevilla slowly found their rhythm and went ahead when Adriano picked up a short corner from Navas before sending a cross to the far post where Squillaci headed home.
That was enough to take the wind out of Stuttgart's sails and reduce them to the mistake-prone team that stands a lowly 13th in the Bundesliga.
Navas killed off all hope for Stuttgart when he stabbed in from close range after former Germany keeper Lehmann failed to clear a cross, sending it into his path just after the break.
Frederic Kanoute should have made it 3-0 when he had only Lehmann to beat but delayed and allowed Boulahrouz to clear.
Squillaci was on target for the visitors, though, when he again headed into the net at the far post.
Substitute Elson almost immediately pulled a goal back for Stuttgart with a curled freekick and Serdar Tasci could have made it closer but his stoppage-time volley hit the bar.
"We are making far too many silly mistakes," said Stuttgart coach Markus Babbel, who is finding things much harder in his second season at the club.
"But the result tonight and the performance do not match. I think we had a good game."
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