After a goalless opening period, in which Kevin Mirallas twice struck the woodwork, the Potters took the lead on 49 minutes via a strike by on-loan Liverpool winger Oussama Assaidi.
But Everton were not to be denied and pressed forward strongly in search of an equaliser.
After Steven Naismith and Mirallas had seen efforts blocked by Erik Pieters, their goal duly arrived thanks to Leighton Baines' injury time penalty after Jermaine Pennant had fouled fellow substitute Leon Osman in the box.
Despite earning a point, the Toffees dropped from fourth to fifth in the table - being leapfrogged by arch-rivals Liverpool, who beat Hull City - and Martinez had wanted to win the game.
He told Sky Sports: "We feel disappointed in terms of we see it as dropping two points today.
"But as you know in football, we knew that Stoke have got a very good record at home.
"They are unbeaten in the last six (league games prior to Wednesday's clash), they won their last two, and we knew it was going to difficult, especially if you don't get that early goal or the first goal.
"But given the circumstances, when you get an equaliser that late and you've got two balls cleared off the line, you feel maybe it's not going to be your day.
"When you still get a positive result away from home, you have to take it and move forward."