The Merseyside club shared a 2-2 draw with Norwich City on the opening day, and were held to a 0-0 stalemate by West Brom at Goodison Park on Saturday.
But Martinez believes his team deserved more from both fixtures, and has argued that their performances bode well for the future.
"I'm very encouraged. I think in both draws we were the team that deserved three points," he said.
"I think in any draw you look at it and you think 'we got away with a point' or 'we were unfortunate not to get the three points'. I think at Norwich clearly with the amount of chances that we had and scoring the second goal before the home side it was a disappointment to drop the two points.
"Today, I thought the first half we got a little bit frustrated but as the game went on you could only see us scoring. Yes, West Brom have got a real good threat in set plays and we dealt with it really well, they were really well poised in the counter-attack and I thought we coped with that really well.
"There are many areas we need to improve but keeping a clean sheet today in the manner that the game was is very encouraging and I see two positive draws rather than points dropped on both."
The Spaniard handed a second consecutive start to 19-year-old midfielder Ross Barkley for the visit of West Brom, and he has hailed the youngster’s early contributions.
"I've been very impressed with Ross. I think at his age it's incredible the manner that he has developed his game," Martinez added.
"Sometimes you see a youngster with enormous potential and enormous talent and somehow you need to use that. With Ross it's the opposite.
"He can fit in in different roles, he's a thinker, he makes fantastic decisions in where the space is. I've been very impressed so far and I know he's going to go from strength to strength, he's an incredible talent."
