Woltemade put the visitors in front with a header after 31 minutes, scoring his first goal in Germany colours.
“It was a very important three points for us,” Woltemade told Germany’s RTL, adding “in this atmosphere and in a bit of an ugly game, you have to win and we did well”.
Germany coach Julian Nagelsmann agreed the win was “ugly”, adding “it definitely wasn’t our nicest game — the stadium was extremely emotional and we had to fight for every ball”.
The win kept Germany ahead on goal difference of Slovakia, who beat Luxembourg 2-0 on Monday.
Both Germany and Slovakia have three wins in four matches, with only the first-placed side guaranteed to qualify for next year’s World Cup. The Germans host Slovakia in Leipzig in November.
Despite the loss, Northern Ireland are only three points off the lead. Their final two fixtures are away against Slovakia and at home to Luxembourg, who have lost four from four.
Looking to qualify for a World Cup for the first time since 1986, Northern Ireland came into the match with wind in their sails after a shock 2-0 win over the previously unbeaten Slovakia.
Nagelsmann also wanted to keep the momentum from Friday’s 4-0 win over Luxembourg, naming the same starting team two games in a row for the first time since Euro 2024.
The Germans survived an early scare when Daniel Ballard’s 14th-minute strike was rubbed out for an offside in the build-up.
Just past the half-hour mark, Woltemade headed in a David Raum corner to put Germany in front.
Only one of Woltemade’s 17 goals for Stuttgart last season were headers but the 1.98-metre striker has clearly worked hard to make the most of his height advantage.
Since making his big-money move to Newcastle in the summer, three of Woltemade’s five goals for club and country have come from his head.
Karim Adeyemi had a golden chance to double Germany’s lead just after the break when put through on goal by Florian Wirtz, but the Borussia Dortmund forward hit his shot wide of the post.
Northern Ireland forced a corner in the hope of pinching a point in stoppage time but the Germans held firm, Bayern Munich defender Jonathan Tah clearing the ball away.