At the final whistle, Frank was seen being snubbed by Spurs pair Micky van de Ven and Djed Spence as he urged them to not walk off the pitch before applauding the home support.
Instead, both players stormed away and also resisted the efforts of coach Andreas Georgson not to ignore the manager.
Asked about the blow-up, Poyet told Tribalfootball.com: "It's a risk that now the coaches in the last two or three years they're taking getting onto the pitch after the game. It didn't happen previously.
"The coaches will come together, shake hands, get in. Then from two or three years now, the coaches after they shake hands, they go onto the pitch, they go into the middle of the park, sometimes you shake hands with the referees.
"And where there is tension, any game there can be tension, especially in a derby, you're going to go into where every player is having a reaction. You can have a war with an opposite player that may be upset because you beat them, or with one of your players, or with the referees as well. So I think it's an unnecessary risk that now the coaches are taking it because it could become a habit.
"Imagine, a derby, you know that it's a must-win game, you're playing at home, you didn't play well, you are frustrated. And on top of that, when you finish the game, the fans, they boo you massively. And when you want to go in because everything is bad, it's dark, the coach wants to say something, whatever it is, you know, and you don't want to hear anything. You just want to disappear."
Frank must be smarter
Indeed, Poyet says Frank's actions did contribute to the controversy by his wish to enter the pitch.
He further explained: "It is very individual, you know. Not all the players are the same and react the same to a game like this one. So, like I said, look, why I don't like the VAR? Because we're making the referees be the centre of the game. You know, the camera follows the referee. They wait for him to talk now. They are like actors. I hate it. The referee, the best referees in the world, they are when nobody knows who refereed the game because there were no issues.
"When you know the name, when you know the face, it's been too much in VAR. You know, I don't like that. And now the coaches are doing the same. They finish the game, they go in. Where the cameras go? Before they would go to the scorer, the one who scored the winning goal.
"Now they go to the coaches and they follow the coaches on the pitch. So they're becoming like the focus of the television. And when you have too much of the focus on you, whatever it is, the little things, boom, you're going to be the next day, everybody asking you about that."
- Gus Poyet was speaking to Tribalfootball on behalf of BetWright