The Serie A leaders won 2-0 in Germany in the first leg after goals from Victor Osimhen and Giovanni Di Lorenzo.
Eintracht were reduced to 10 men early in the second half when Randal Kolo Muani was dismissed for roughing.
“It was an end-to-end game, which we want and thrive in. We need to stay calm, very calm,” Spalletti told Sky Sports.
“There’s the second leg still. Our biggest enemy is to think it’s done We have proved that we belong there, but we still need to earn it in the second leg.”
Napoli also missed a penalty in the first half, with Khvicha Kvaratskhelia spurning the opportunity.
Despite that, Spalletti was delighted with Napoli’s performance.
“I am very satisfied with the performance, because from the start the team played with character, took control of the game and did what we wanted to do,” he said.
“There are no complaints, although we had the opportunity to score more goals, but this is good.
“The key was that in trying to score a second goal, we must never lose our balance. The team took a mature approach and I am pleased with that.
“The dangerous thing here was allowing them to take control of the game, as they cover spaces that tactically with the 4-3-3 tend to empty out.
“My team did really well to press high, refuse to let them start those counter-attacks, and that immediate aggression was crucial to prevent them getting back into shape.”