First-half own-goals from Antolin Alcaraz and Seamus Coleman in Saturday's Premier League clash at St Mary's ensured the Merseyside club remain a point behind fourth-placed Arsenal, who play Newcastle United on Monday, in the race for a UEFA Champions League spot.
Everton did respond well in the second half and had two separate appeals for penalties waved away by referee Oliver.
On the first occasion, substitute Osman was booked for diving after going down under a challenge from Dejan Lovren, before the Southampton defender was perhaps fortunate to avoid conceding a spot-kick following a tangle with James McCarthy.
Manager Martinez expressed his disappointment at Everton's misfortune and was puzzled by Osman's caution after his side's Champions League hopes suffered a major setback ahead of the final two games of the season.
"Conceding an own-goal so early was a massive blow," Martinez said. "Then to concede again from such a cheap action, psychologically it did affect our performance.
"We are very young group, we'll learn from today. If you concede two goals and you've got 60 or 70 minutes to go, it's long enough to get something out of the game. We need to learn not to get frustrated and not to get affected by it.
"I think the second half we needed that little bit of luck. Leon Osman for me it is very clear, that when a centre half goes to ground he has to get the ball, otherwise as an attacker or a creative player he'll have to been impeded.
"It was a cheap decision to book the player, I'm sure that the referee was looking at Osman, and he didn't realise that the centre-half just threw himself to ground without touching the ball. For me that should have been given as a penalty.
"The other one is a little bit of a tug. It could go both ways."
Everton face title-contenders Manchester City at Goodison Park and a trip Hull City in their two remaining top-flight games.
