Martin Olsson's stoppage-time strike secured back-to-back victories after Aleksandar Mitrovic twice drew the Magpies level.
Swiss defender Klose, a January signing from Wolfsburg, had opened the scoring for Alex Neil's men, who are now four points clear of Sunderland and six better off than Newcastle, although both have played one game fewer.
Norwich can draw level with 16th-placed Crystal Palace by winning at Selhurst Park next Saturday and play Sunderland at home the following week in two games key for their survival.
"Some games before we've not had luck on our side," Klose told Norwich City TV.
"This time we did, and it felt fantastic to win the game. I'm very happy at the moment because we've got two big wins in our last two games.
"We've got more big games coming up. We want to take points from those as well."
Olsson's strike was his first since September 2014.
"It's a really good win for us," the Sweden left-back said.
"We were thinking about this game ahead of the international break.
"It was good to get out on the pitch and get the three points.
"The only thing we had in our mind was winning. We didn't want to draw so we went for it at the end and got the luck.
"I hardly ever score, so it was a great feeling. I didn't know how to celebrate so I just ran around like a headless chicken.
"We have to keep playing well in order to keep this going."
Newcastle play Southampton, Swansea, Manchester City, Liverpool, Crystal Palace, Aston Villa and Tottenham in their final seven games.
Boss Rafael Benitez maintains survival is possible, if Newcastle can show the same level of performance as they did in the second half at Carrow Road.
He struggled to explain the insipid first-half display.
"For 10, 15 minutes we were fine," Benitez said.
"After we made one or two mistakes, then we gave them the belief they could win.
"We had a bit of anxiety, some nerves, made three, four mistakes.
"When you are at the bottom of the table you have to manage the situations and you have to manage your character, your nerves.
"We didn't do well in the first half and (did) much better in the second half."