Debutant Nico González set the tone from midfield, playing a raking pass into Amar Dedic, whose header found Elanga to turn away from Sandro Tonali and finish low into the corner in the 62nd minute.
Wissa added a second before the close to complete Newcastle United's first league win of the campaign and leave Roberto De Zerbi's Tottenham bottom of the table without a goal to their name.
Jaissle could barely contain his satisfaction afterwards.
"Incredible. Really proud, we are really proud as a club and coaching staff to see the team delivering on the pitch and also seeing players committed to what we are doing," he said. "We did well also last week.
"It was a step in the right direction again with a strong opponent, huge signings. The way we handled the opponent was amazing. It is always the same as a coach, you try to emphasise where you can stretch the opponent and find gaps. We tried to keep our principles.
"The DNA of Newcastle is a good fit to the playing style I like to see so therefore we saw already in the past similarities."
On González's contribution, he added: "I am happy that you somehow think I am smart. When he has the ball he has a good timing, when to deliver and also when to slow down the game. A really good debut."
Asked about the progress of Anthony Elanga and Yoane Wissa under his coaching, Jaissle was keen to spread credit.
"I don't take the compliments yet. I try just to encourage them, motivate them and support them, not only those two but the whole squad. I am really happy to see the improvement. It means a lot. I just said to the lads in the locker room we can count now our defensive stability. The clean sheet today was a big goal."
For De Zerbi, it was a familiar story of control without end product.
"I think for 65 minutes we played better than Newcastle, we had a chance to score and didn't score. And then after the goal we lost the distances," he said. "We had more control of the ball, we can improve in the defensive phase with the distances. It is part of this work when you change many players and start a new project with a new team."
The Tottenham boss urged patience despite the poor start.
"We will not panic. I know very well where we are going, what the plan is. I am sorry for the fans and sorry for the players for the result but we have to improve, and to change two seasons like the last two is impossible in 10 days and with people in the transfer market. Now we have to keep calm and focus on improvement. If the rush is more important than the journey we can't find the solutions to the last two seasons."