Leonardo Jardim's Olympiakos made it five wins from five courtesy of Spanish midfielder David Fuster's first-half goal at the Peristeri Stadium on Saturday, keeping them three points ahead of Panionios who defeated AEK 1-0 on Sunday thanks to a 16th-minute strike by Leonidas Kabantais.
That result spelled the end of Vlachos's tenure at AEK, with the club confirming his departure via a brief media statement on Sunday.
The 50-year-old former AEK player presided over one of the worst starts to a season in the club's history with the Yellows losing four of their opening five games.
AEK are bottom of the table with just a single point, following an off-season of financial uncertainty in which they sold virtually their entire first-team squad.
Former PAOK Thessaloniki and Greece coach Angelos Anastasiadis has emerged as the early favourite among local media to succeed Vlachos.
Panathinaikos's indifferent start to the season continued on Sunday as Jesualdo Ferreira's team were held to a goalless home draw by Asteras Tripolis, a result which prompted a large section of fans to jeer the under-fire Portuguese coach off the field at the Olympic Stadium.
"Neither the fans, nor ourselves are satisfied with the current situation and the reaction is understandable," Ferreira told reporters afterwards.
"This is the way it goes in football sometimes and we can only change things through hard work. It's imperative to get a win in our next match if only to bring a smile to the players' faces."