Everton were far from their best against Gus Poyet's relegation-threatened side and needed a Wes Brown own goal to make it seven league wins in a row.
In a game of few chances, Brown's mistake 15 minutes from time kept Sunderland deep in relegation danger and ensured Everton moved two points clear of Arsenal into fourth.
Martinez felt it was always going to be difficult to match their 3-0 win over Arsene Wenger's men last weekend and was simply relieved to secure the victory.
"It's really, really satisfying because we knew it was going to be a difficult game. I thought it was very tense, very cagey with not a lot of quality," he said.
"I think the emotions (relating to) the significance of the three points for both teams was a part of the performance and I was always worried that, off the back of such a stylish performance (against Arsenal), it was a very dangerous ground we were coming to.
"I thought we got through that with flying colours. I don't think we created too much going forward, maybe just three chances, but I'm glad we took one of them and got three very important points in a very testing environment."
With Everton's European fate in their own hands, Martinez's men have every opportunity to make certain their first UEFA Champions League appearance since the 2005-06 season.
The Spaniard claims the appetite within the squad will help them as he and the players enter unchartered territory.
"(Staying in the top four is) our objective, we're going into new ground. We're going to take every game with a real excitement," he continued.
"We've got a group of players who are focused and desperate to play, with good competition for places.
"We're just going to take it step by step, but to have 66 points by this stage of the season is a remarkable achievement and the players deserve a lot of credit."
