Goalmouth action was scarce in a tense affair at Goodison Park, Liverpool's Jordon Ibe striking the post in the first half before Everton's Seamus Coleman brought a smart save out of Simon Mignolet late on.
After the 224th edition of the derby ended all square, Martinez believes his side can approach the rest of the season with optimism.
"We've felt that [the season was turning around] for a while now," he told Sky Sports.
"It's another clean sheet and we're starting to enjoy our football.
"We're getting players back from injury and we're going to get strong when everyone is back.
"Liverpool were in a very good moment of form. If you make a mistake they can hurt you.
"The game was about making sure a team in great confidence couldn't get in a position to dominate the game.
"When we got on the ball we created good space and got in good positions to hurt them.
"We're really looking forward to the next part of the season."
