Media playback is not supported on this device
Goals from Divock Origi, Mamadou Sakho, Daniel Sturridge and Philippe Coutinho gave the hosts victory over the Toffees, who had Ramiro Funes Mori sent off five minutes into the second half.
Martinez, under pressure after a run of just three wins in 10 games, said he was "embarrassed" by the performance.
"It doesn't come any worse than that," he added.
"Everything that could have gone wrong went wrong. The whole chain of events was horrific and it is a showing we need to forget as quickly as we can."
Everton have not won at Anfield since 1999 and, after conceding two quick goals before half-time, they never looked like taking the three points on Wednesday.
In the end the Reds had 37 shots on goal compared with just three for Everton, none of which were on target.
"We know we shouldn't have put a show on like this," said Martinez.
"Sometimes there are things in football that you can't control with events that go against you, but we brought that on to ourselves conceding those two goals in three minutes just before half-time and it's something we should be much better at."
Martinez faces the task of picking up his players for Saturday's FA Cup semi-final with Manchester United at Wembley.
Louis van Gaal's side have won five of their past six games, while Everton have not won since beating Chelsea in the quarter-finals on 12 March.
"I think the game is so big at Wembley that it is a game they don't need lifting," added Martinez.
"It's tough but we're going to face adversity as we have to do, as we always show with strong character, with strong responsibility.
"We're going to make sure we make up for this performance today in front of our fans at Wembley."
Martinez will be without defender Funes Mori for that game following his red card for standing on Origi's ankle.
"Ramiro tries to win the ball. That is his intent - he doesn't try to hurt the player," said Martinez.
"It is a mistimed challenge and at that time he gets Divock Origi's ankle. It could have been a yellow card, it could have been a red card. I can understand why the referee gave it.
"Ramiro probably got carried away with the emotion of the derby."
There could be a further concern for Martinez after fellow defender John Stones was substituted against Liverpool with a stomach problem.
The Toffees finished the game with midfielders James McCarthy and Muhamed Besic in the centre of defence.
"I don't like criticising managers as we have all been there but Roberto Martinez doesn't seem to have a plan B," former Everton midfielder Peter Reid told BBC Radio 5 live.
"He goes 4-2-3-1 but you've got to have something else. People work you out."
Never want to miss the latest Everton news? You can now add them and all the other sports and teams you follow to your personalised My Sport home.

Timothy Smith, 57, was pushed during the robbery in The Drive, Hullbridge, at about 23:00 GMT on 19 March.
He fell heavily on to a low level wall and suffered a serious brain injury. He died two days later.
A 50-year-old man from Southend has been detained on suspicion of murder, robbery and possessing a firearm.
Detectives are appealing for anyone who saw a Ford transit van in or near Ferry Road, or two men at or near a bus stop on Ferry Road in Hullbridge on 19 March between 19:00 and 23:00 GMT, to contact them.
Everton boss Roberto Martinez described Wednesday's 4-0 defeat at Liverpool as his worst moment at the club.
[NEXT_CONCEPT]
A man has been arrested on suspicion of committing a murder during an armed robbery in Essex.