Liverpool's ninth league defeat of the season came just days after a morale-boosting victory over Bolton Wanderers and leaves Hodgson anxiously fretting over his future.
"It was a real shooting in the foot exercise," Hodgson, who is the bookmakers' favourite to become the next Premier League manager to lose his job, told Sky Sports.
"It was very difficult to go in two goals down at halftime and we compounded that by conceding an equally poor third goal."
Steven Gerrard, playing his 550th game for Liverpool, pulled a goal back for Liverpool in the second half after two goals by Benjani and one for Martin Olsson had put Rovers in command.
Gerrard also missed from the penalty spot to complete a miserable night for Liverpool who are in 12th place, just four points above the relegation zone.
"It's a bitter blow for us and for the fans and we are all very, very disappointed and frustrated," added Hodgson, whose side face Manchester United in the FA Cup this weekend.
"We are very dejected. The fans supported us well and I can understand how bitterly disappointed they must feel but all I can say to them is that if they saw our dressing room they would see similar dejection."
