While Cartagena took an early lead on Friday, Martino's side knocked off the third-tier club 4-1 to end their two-game losing streak in all competitions.
Barcelona lost to Ajax Amsterdam in the UEFA Champions League and Athletic Bilbao in La Liga in their previous two games but bounced back against Cartagena to set themselves up for a spot in the final eight of the Copa del Rey.
Martino was impressed with how his team responded after a tough week.
"We were obliged to win, bearing in mind our recent outings, and we did," Martino told AS after the match.
The 51-year-old Argentine praised Cartagena for the way they began the first leg but argued the hosts' energetic start, which saw them take the lead in the 16th minute through Fernando, left them without enough energy to finish off the contest.
"They played a very intense game, they did a lot of running and their attitude was good during the opening 10 or 15 minutes," Martino said.
"Later, when we had control of the ball, they had no energy left."
Martino was also thoroughly pleased with the performance of Jordi Alba as the 24-year-old full-back played his first game since September.
"We had planned for him to play less but in the end, he lasted about 80 minutes without any problems," he said.
"We need to make sure he feels confident when he strikes the ball.
"We're all very happy with him. The situation couldn't be better for the second half of the campaign."
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