Crook, a former Tottenham Hotspur and Norwich City midfielder, stepped down with the club languishing in eighth place in the 10-team A-League with just two wins this season.
The 49-year-old Englishman had been on shaky ground after his team were thrashed 7-2 at Central Coast Mariners last week.
A 3-2 home loss to bitter rivals Melbourne Victory on Saturday, after they led 2-0 in the final 15 minutes, compounded the pressure.
Crook, who secured the job in May after former Socceroos coach Graham Arnold turned it down, said the pressure had become too much to bear.
"It builds up. All this (media) has become a bit of a burden," he said. "With Alessandro coming, the whole thing has become a bit bigger.
"It's become a bit of a weight on my shoulders, to be honest with you, and I haven't handled it that well."
Del Piero missed Sydney's loss to the Mariners due to a hamstring complaint but was welcomed back for the Victory match and the 38-year-old World Cup winner was influential in setting up both his team's goals.
But Crook substituted the former Juventus player in the 66th minute, a move that dismayed the Sydney crowd and saw the coach involved in a heated exchange with a clutch of angry fans after the final whistle.
"I don't regret taking it on ... but I don't think I'm the right person," Crook said.
"It just became a really heavy burden, where it was getting to the stage where I just wasn't feeling well. And my health and my family are the two most important things."
Assistant coach Steve Corica will step in as caretaker until a replacement is found.
Sydney were inaugural A-League champions in 2004-05 and have styled themselves as the fledgling league's glamour club.
"We have a lot of expectations. I can sense (the fans') frustrations,” Sydney FC chief Tony Pignata said.
"We ask for their patience, I know some won't be patient."