The Magpies earned a point in a 0-0 draw at West Ham on Saturday but are now just three points above the drop zone following Wigan's win at West Brom on the same day.
Following recent criticism, Gutierrez was more concerned with the encouraging togetherness Newcastle displayed at the weekend rather than results elsewhere.
"I think we showed that was a performance of a united, committed team," said the Argentina international.
"Everybody could see we were fighting for each other. It has been a difficult few weeks and we haven't been solid or compact but today we were.
"We walked off the pitch and heard the other results but really I don't care about the other teams, I'm only thinking about Newcastle and what we have to do. This result was a good step in staying up.
"Away from home we haven't had many good performances this year, now we look forward to the next game and hopefully the same kind of performance can get us a win.
"No one who was here could say we weren't playing for each other, fighting until the final whistle.
"If we'd played more games like this in recent weeks I don't think we'd be talking about a relegation battle.
"We showed the fighting qualities we need to survive. If we do the same things I think we'll be OK. Fight, fight, fight really hard, work for the team and try not to concede.
"It was important not to lose this, even more so to keep a clean sheet and now there is another big game. We know we must get the three points next week.
"We are all in this fight together. We all want the same thing."