Gotze's fine control and shot sunk Lionel Messi and company and earned Germany their fourth world title.
"It is unbelievable," Neuer, who won the Golden Glove award for the best goalkeeper in the tournament, was quoted as saying by kicker.de
"The team has done superbly, not only the players, but also the team behind the team. At some point we will stop celebrating, but we will always stand up again with a smile."
Captain Philipp Lahm added: "What we have done and how we have worked is incredible."
Lahm, consistently excellent for Germany all tournament, added: "Whether we have the best individual players or whatever does not matter, you have to have the best team.
"We stepped up time and again in the tournament, did not let ourselves get distracted by any disruption, went on our way.
"And at the end you stand there as world champions - an unbelievable feeling. The team has remained quiet and patient."
Centre-back Mats Hummels had his work cut out against Messi and company.
"I am still completely trapped in another world, physically too exhausted to be able to rejoice euphorically. But that will come in the coming days," he said.
"You can only be world champions when you act as a team. We wanted to make that clear to everyone the whole time and we achieved that, even if we got that little bit of luck in the final as we were already a bit waiting for penalties."
Christoph Kramer, the midfielder who came in as a last-minute replacement for Sami Khedira only to have to go off in the first half, clearly very groggy, after a heavy collision with Ezequiel Garay, said: "I can't remember very much, but that doesn't matter now."
Argentina coach Alejandro Sabella admitted that his side had paid the price for missing chances as Germany were crowned World Cup winners but said he was proud of his players.
Lionel Messi and Gonzalo Higuain were both guilty of being wasteful as Germany triumphed 1-0 in extra-time.
Sabella said: "The players are bitter and sad because we had a huge dream having made it all the way to the final. We needed to be more clinical and I am sad just like the players.
"Beyond the pain of the defeat in the final they can look themselves in the eyes and say they gave everything for Argentina.
"We have an extraordinary group of players, these were warriors. I congratulated the players because they left every drop of sweat out on the field. They gave their all.
"I am very proud, the boys played an extraordinary World Cup and improved as they advanced through the stages and they gave everything for the Argentina jersey.
"I think that this was a very tough match, Germany started better and handled the ball well and is a great team.
"The match had its up and downs but we had the clearer chances. I congratulate my players for their extraordinary work and congratulations to Germany on the title."
Sabella also refused to be drawn on his future.
He said: "I cannot speak about the future, the future for me now is to be with the players and my people - the coaches and my family - and to rest a bit. I don't know what to say about the future."