Fletcher came on at half-time at Cardiff on Saturday and notched a late strike as the Black Cats came from two goals down to earn a point.
The draw made it four games unbeaten in the Premier League for Gus Poyet's side as they attempt to become the first team since West Brom in 2004/05 to be bottom on Christmas Day but stay up.
The Black Cats currently sit just two points from safety and Fletcher - who suffered relegation with Burnley in 2009-10 and Wolves two seasons ago - believes they still have a 'massive chance' to avoid the drop.
He said: "Compared to the other two I've been in, this is probably the tightest one.
"At Burnley I think we were dead and buried by this time.
"We've still got a massive chance. Our home form's been good, we've beaten Manchester City and a few other teams, so we've just got to keep it going.
"It's very tight down there. There's only a few points in it so we're not adrift. If we can keep picking up points like this then we'll be fine."
The goal was Fletcher's first since October and the striker hopes to have pressed his claims for a starting berth against Aston Villa on New Year's Day, where he could line up alongside fellow forward Jozy Altidore.
He added: "I've not scored for a while so it was nice to get on the scoresheet.
"It was good to show the gaffer that me and Jozy can play together. Hopefully he takes that on board and we can get together on Wednesday."