Villa leapfrogged Spurs into the top four, as goals from Pau Torres and Ollie Watkins earned the visitors a 2-1 comeback win after Giovani Lo Celso opened the scoring.
Depleted Spurs, who were without 10 players and lost Rodrigo Bentancur to another injury in the first half, had three Heung-min Son goals ruled out for offside, twice hit the post and forced a string of saves from Emi Martinez.
Villa also had a Watkins goal disallowed for offside and struck the woodwork in an entertaining contest.
We need to have a positive energy... we have to improve only mentally and we'll win in the future if we do this
Spurs's latest reverse follows defeats by Chelsea and Wolves, and next up is a daunting trip to Manchester City on Sunday, but Kulusevski insists they will return to winning ways if they improve mentally and stick to Ange Postecoglou's approach.
He said: "I really hope [the fans] stay with us because we will win in the future if we continue [to play] like this.
"It is important that we stay together, everybody in this club, and outside is not important what anybody else says. Inside, everybody needs to have a positive energy and help us players, because we will bring them happiness.
"We played unbelievable. First 30 minutes [were] unbelievable, fantastic from everybody. When you play like that, you want to win, but this is the way. We have to continue.
"We lost three in a row, that's the big problem. The way we played, especially the first 30 minutes, we have to play like that for 90 minutes — then we cannot lose games. We have to improve only mentally. That's the only thing.
"Football-wise we are unbelievable, the coach is doing unbelievable with us. We have to just stay on track, believe in ourselves and keep playing like this. It's small details. We should be up a couple of goals. If we are more clinical there would be smiles after the game."
With Pape Sarr, James Maddison and the suspended Yves Bissouma all absent, Kulusevski moved to No8 alongside Lo Celso — making his first League start since Nuno Espirito Santo's final game in charge in October 2021 — and Bentancur, who returned to the XI for the first time since ACL surgery in February.
The Swede has previously said his best position is No10 and he impressed centrally, striking the post early on and contributing to a number of incisive moves.
"I felt so good, I felt like the real Dejan," he said. "I felt like myself. The coach decided where to put me. I enjoyed it and I had real fun playing. It was a great day for me.
"I've done it [played centrally] all my life, so it's nothing new to me. Of course, in the second half I played right wing, but physically I feel really good, so I'm happy and I know I'm on the right path."
He added: "Lo Celso was so good and he's going to help us. He showed what kind of player he is and I expect him to play a lot more now."
Rodri was really angry and he said it could maybe be more than a yellow card.
Bentancur was forced off after just half an hour following a nasty challenge from Matty Cash, earning the defender a booking and prompting a furious reaction from several Spurs players.
"Rodri was really angry and he said it could maybe be more than a yellow card," Kulusevski said. "It's just hard for Rodri. Unbelievable person, unbelievable player. Out for eight months and comes back after two games and gets injured again. Life is hard. He has to stay strong and he has a great family to support him and good team-mates.
"It's very hard [to see] because I love him as a guy. He's the guy I want to go to war with and they take him out like that with a bad tackle. That's life."
Spurs's injury crisis is not expected to ease before the visit to the Etihad, although Bissouma will be available again after suspension.
In spite of their absences, Kulusevski's says Sunday's performance should give Spurs confidence they can get a result at the home of the champions.
"Absolutely. We played there a couple of times before with Antonio [Conte] and did good, so it's going to be a good game," he said. "City's a great team, but we're a very good team also. I love playing against them. I've played them twice and scored two goals. I'm looking forward to doing it again."