Magpies goalkeeping coach Andy Woodman and Saints counterpart Toni Jimenez were both sent to the stands after a pushing match between the two benches following substitute Morgan Schneiderlin's rugged injury-time challenge on Massadio Haidara.
Pochettino said: "What happened at the end of the game is not a good example, it's not good for football. From our point of view, we are going to try our best not to let it happen again because it's not good for anyone and not good for the sport.
"I think it is avoidable but also in a high-intensity game such as that, any little spark can lead to something else and I think that's what happened.
"From our point of view, it's something that we regret and that we don't want ever to happen again.
"Let's stay calm, we need to be calm about what happened today. These things can happen and we need to keep it indoors, not publicise it and make a big thing out of it."
Opposite number Alan Pardew insisted the incident should not be allowed to overshadow a hard-fought encounter which both teams had chances to win.
Pardew said: "It was a bit of panto season. I'm not going to get carried away with that because it shouldn't take the headlines.
"The battle is on the pitch and that was a terrific battle. The one on the sideline was pretty pathetic, what I saw of it, and best forgotten.
"I had my back to it most of the time. I just said to their manager, 'Just get your staff in, I'll get my staff in and let's get on with it because we want to win the game', and he was of the same opinion as me."
Newcastle took with lead with 28 minutes gone when Yoan Gouffran reacted first to Mike Williamson's knock-down to round keeper Paulo Gazzaniga and slide the ball into the empty net for his fourth goal in as many games at St James' Park.
But the Magpies failed to build upon their lead and with the Saints a much more dangerous proposition after the half-time introduction of Adam Lallana, they drew level with 25 minutes remaining.
Jay Rodriguez exchanged passes with Steven Davis, who had hit the post seconds earlier, before coolly beating Tim Krul.
Both sides went for a winner as time ran down but in a stormy conclusion, referee Mike Jones was left with a bloody nose after being caught accidentally by Magpies midfielder Moussa Sissoko.
Pardew said: "The ref has got a bloody nose, he is trying to patch himself up for the Christmas party.
"But he took it in good heart, really, I have to say. He realised it was a little bit of a flick out at the goalie, who had flicked out at him [Sissoko].
"It was all a bit silly, but unfortunately he ended up with a bloody nose."
The Saints were indebted to the impetus provided by Lallana, who did not make the starting line-up because of a niggling injury.
Pochettino said: "Adam had a slight injury and he wasn't fully fit to be playing the full 90 minutes.
"We are not going to take any risks with our players, so we thought he could come on in the second half as a sub, come off the bench and have an impact, and we are delighted at the way he did."
However, Pardew was left ruing two glaring misses by striker Loic Remy, who has now gone four games without a goal.
He said: "We had two big chances today that fell to Loic. I felt he played well today but those two chances, I expect him to take and that can be the difference in a game that is as tight as that.
"It probably cost us today."