The Toffees found themselves behind to a James Collins header at Goodison Park in the Third Round tie and looked to be on course for a fifth straight defeat.
Everton 1-1 West Ham United
The FA Cup
Third Round Proper
Tuesday 6 January 2015
Goodison Park, Everton FC
Live on BT Sport
Winners receive £67,500 from prize fund
But a stoppage-time goal from Romelu Lukaku rescued a 1-1 draw to set up a replay at the Boleyn Ground next week and leave Martinez relieved at the result.
"It would have been very damaging but you always get an opportunity to win the next game," said Martinez.
"At the moment, The Cup is such an important competition for us so it would have been a horrendous setback.
"It is very important we are still in The Cup and now look forward to the next game in the league [at home to Manchester City] but we are still aware we are in a bad run."
Martinez is no stranger to success in The FA Cup, after winning the competition with Wigan Athletic in 2013 and taking Everton to the quarter final stage last season.
And he revealed there was a real feeling of relief in the Everton dressing room after the game, with the importance of Lukaku’s goal described as "huge" by the Spaniard.
"When you walk into a dressing room with a defeat it is a completely different situation," he explained.
"It is a big moment to be able to go in with different emotions and I am extremely proud of the players.
"In a period like ours you have two options, you can shy away and hide or you can take responsibility."
