The Black Cats looked certainties for relegation just a few weeks ago but have salvaged the situation with a superb run of results against some of the Premier League's top sides.
Their latest tour de force came at Old Trafford on Saturday when a first-half Seb Larsson goal earned Sunderland a 1-0 victory over Manchester United.
And Mannone said: "It's a great achievement from the club and a massive three points. We are in a really good position now.
"We played great, created chances, hit the crossbar and the post. If one of them had gone in, there might have been a bit less tension at the end!
"I thought we defended really well and Manchester United did not have many chances."
The win over United followed a 2-2 draw at leaders Manchester City, a stunning 2-1 victory at Chelsea and a 4-0 thrashing of Cardiff in a relegation six-pointer.
Mannone added: "We started playing well and sometimes playing away against the big clubs can help you, rather than kill you. It was like that for us this season at least!
"We went into the City and Chelsea games with the mentality of trying to get the three points and we did something incredible.
"We lost our form a little bit after the (Capital One Cup) final but, since the manager came in, we changed our mentality, we played a bit more football, and we believed in our quality again.
"We have really been solid lately. I didn't have much to do in the Cardiff game either and that shows everyone is doing their job.
"We have been soft at times this season and I was bit busier then but every player has been fantastic in the past few weeks and I think we just need to carry on now.
"Hopefully, relegation will be gone and we can play with a little more freedom and without tension."