James Collins gave the visitors the lead in the 56th minute of the tie at Goodison Park, the centre-back heading in from Morgan Amalfitano's corner.
However, striker Romelu Lukaku fired home an equaliser in the first minute of added time to force a replay and halt a run of four straight defeats for the Merseysiders.
And Martinez believes the result was just reward for a strong display of character.
"It would have been a crime if we had been out of the cup," the Spaniard told BT Sport.
"I am pleased with the character and the reaction we showed. We are low on confidence, but it was a phenomenal reaction and it could easily be a turning point.
"First half we created the best opportunities and didn't take them. Then, we were open to conceding from set-pieces.
"But the character we showed, when we're not in a great mood at the moment, was of a huge satisfaction.
"The replay was a huge reward and we look forward to that. The late equaliser could be a turning point."
The two sides will meet again on January 13, with the winners facing a trip to Doncaster Rovers or Bristol City in round four.
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