The League One side took the lead through Luke Freeman in the 36th minute, but Gerard Deulofeu's reply just before half time and Marouane Fellaini's extra-time winner sent Boro fans home with broken hearts.
"We deserved to go through," Martinez told reporters. "Marouane Fellaini gave us the push we needed but the headlines should be about how well Stevenage played.
"It was a typical cup tie. Stevenage were really really good, a well-organised team.
"I thought we were very good in two thirds, but in the final third we were at times naive. At times we didn't show enough sharpness."
Everton will travel to Fulham in the third round.