The Toffees boss has come under fire this term with his side having won just four games at home, while they lie 11th in the Premier League.
But they remain just six points behind fifth-placed Manchester United and are now in the quarter-finals of the FA Cup.
Martinez feels Everton are coming into form at just the right time to enjoy a successful end to the campaign.
“Another clean sheet keeps the consistency from our last four performances,” he said.
“We look a really strong team now going into the final part of the season and I am really proud of going into the quarter-finals of the FA Cup.
“We will take it step by step. It was important that we showed we can adapt to this game and now we want to get a really good period in the league. The only thing that has been missing has been wins at home and we are looking to change that.”
Second-half goals from Ross Barkley and Romelu Lukaku sealed Everton’s passage into the last eight after Joel Robles had saved Charlie Daniels’ first-half penalty.
Martinez heaped praised on Robles, who kept his place despite Tim Howard recovering from injury. “He was magnificent in the way he controlled the moment, and in the way he enjoys being the goalkeeper of Everton,” said Martinez. “He has been working so hard behind the scenes. Tim has been an incredible influence on him and that showed today.”
Bournemouth will now return to the main task in hand of ensuring they stay in the Premier League.
Veteran defender Sylvain Distin was disappointed to go out to his former club, but took plenty of positives.
“The work rate is really high at the club. We have a good team spirit in the dressing room and everything is there to have a successful season. I don’t think the cup was a distraction and it won’t be easier in the league now we are out. We change the squad in the cup so we saw it as a good opportunity, not a distraction.
“We have an important run-in now and there are some big games for us. We’re disappointed to be out but we have to look ahead.”
Federici, Francis, Elphick, Distin, Daniels, O’Kane, Iturbe (Murray 71), Gosling (Smith 77), MacDonald (Ritchie 64), Stanislas, King.
Wiggins, Boruc, Pugh, Butcher.
Robles, Coleman, Jagielka, Funes Mori, Oviedo, McCarthy, Barry, Lennon, Barkley (Mirallas 81), Cleverley, Lukaku (Niasse 81).
Baines, Stones, Deulofeu, Osman, Howard.