Latics, who face Manchester City in the FA Cup final on Saturday, twice led against the Capital One Cup winners but were made to rue costly errors at the back and remain three points from safety and with just two Premier League games left to play.
Roger Espinoza and James McCarthy struck for the home side, but Martinez's old club responded through Angel Rangel and Itay Schechter, before Dwight Tiendalli hit the winner 14 minutes from time.
Martinez told Sky Sports: "Everything was under control with our first half performance. We were performing really well, but then in the second half with the goal that we give away, from that point on it becomes a very, very difficult game.
"We have so many injuries at the back and today that showed too much. We conceded sloppy goals with bad decisions. That is very unfortunate because you could see the effort of the players and the commitment and desire in wanting to win the game.
"You always get situations in games where you can recover. Emotionally today was such an important game for us, playing at home, that it becomes a difficult performance. When you make a mistake in that manner it becomes a bigger mountain to climb.
"What we must do is make sure we get ready for the next two games that we have got in the league and make sure we get everyone as recovered as we can.
"The team is ready to bounce back and make sure we take the opportunities over the final two games. We have got incredible character and we are determined to fight for every single point that is left.
"There are six points there to fight for and we need to make sure we can learn from this for the final game of the season. We have to accept nothing went our way and we need to make sure we turn it around quickly for the final two games."