A first-half hat-trick from Billy Kee with one from Seamus Conneely and an Adam Dugdale own goal gave League Two Stanley a comfortable win.
Southport, to their credit, kept Accrington out in the second half but Lewer pulled no punches afterwards about his side's performance.
"I thought we were very poor and I'm very upset with the performance," he said.
"We looked lethargic. You've got to give Accrington some credit as well, their movement was very good. Billy Kee and Sean McConville absolutely slaughtered us.
"If I asked one of my players in there, I don't think they could say that they won their own personal battle. If you do that against any team, in any league you will get beat.
"You can't carry eight players in a game, and we were carrying players today."