Teenage debutant Aaron Doran's first professional goal and Mark Carrington's cool finish on the stroke of half-time helped the Dons to a routine 2-0 second-round win.
And Tilson was furious with his side's display at stadium:mk as Southend failed to register a single shot on target in the entire game.
"The one thing I always expect is work-rate and we didn't have that," he said after the game.
"As a club I think we need to apologise to our supporters because we just weren't good enough.
"We were beaten by a much better side and it's very disappointing. A 2-0 loss was the best we could have hoped for and it was only our goalkeeper that stopped it being more."
Source: Team_Talk