Behind at half-time of their home game against Stoke on Monday, Sunderland gained a vital 1-1 draw despite having Craig Gardner sent off before the break.
Belgian international Mignolet believes that the response from the players gives a clear indication that the Black Cats will secure their Premier League future.
"The most important thing is that it's still in our hands," said Mignolet.
"We've got a really big game on Sunday against Southampton and hopefully we can take that second-half display [against Stoke] into that game.
"If the crowd helps us through again we can win the three points and get ourselves safe. It's about getting three points on Sunday and making sure we win, then it depends on what other teams are doing, but we'll see about that after the game.
"We have to deal with our own performances and if everyone puts in a display like we did in the second half against Stoke there will be nothing to worry about."
Sunderland sit in 15th position, three points ahead of Wigan Athletic and with a better goal difference than both Norwich and Newcastle.