"I don't think the performance we produced will bring us serious success if we don't get a handle on things," Rummenigge bluntly told the squad at the post-match banquet.
After three straight wins in Europe, Bayern are on their way to the knock-out stages for a 12th season in a row. However the stuttering away win came at a price as Hernandez suffered a "partial rupture of the inner ligament in his right ankle", the club confirmed Wednesday.
The 2018 World Cup winner, the most expensive signing in Bayern's history, is expected to be out "for weeks" with fellow centre-back Niklas Suele already sidelined for at least six months after tearing knee ligaments on Saturday.
Two goals by Robert Lewandowski turned Bayern's fortunes in Athens on Tuesday after hosts Olympiakos grabbed an early Youssef El-Arabi goal.
Corentin Tolisso scored Bayern's third goal midway through the second half with a curling long-range shot.
However, the German defence looked far from solid, especially after Olympiakos midfielder Guilherme beat goalkeeper Manuel Neuer to set up a tense final 10 minutes.
The performance -- well below that of the dazzling 7-2 rout of Spurs in north London three weeks ago -- also left Bayern's director of sport Hasan Salihamidzic unimpressed.
"You could go crazy watching a game like that. Everything has to be better", he complained to Sky.
For the fifth game in a row, Bayern conceded two goals in Athens, which especially annoyed Rummenigge.
"Again, we conceded two goals," he fumed," we are playing too carelessly - that will lead to trouble at some point."
Bayern top Group B by five points ahead of second-placed Tottenham, who thrashed Red Star Belgrade 5-0 in north London, but the defending German champions are only third in the Bundesliga having dropped five points in their last two league games.
On Saturday, Bayern host giant-killers Union Berlin, who sit 14th in the table yet have already shocked Dortmund and Freiburg in their first Bundesliga season.
Rummenigge has called on Bayern's stars to tackle Union at the Allianz Arena with "the greatest concentration and motivation to score three points" and get back to winning ways after conceding a late equaliser in Saturday's 2-2 draw at Augsburg, which followed a shock 2-1 defeat at home by Hoffenheim.
As Rummenigge pointed out, Bayern can consider themselves lucky, because "despite all the mistakes in the last few games, we are only one point behind" leaders Moenchengladbach " at the top of the table".
"On Saturday, we have the chance to become table leaders, that is the standard FC Bayern has set, we have to live up to that."