Saints fell behind after 36 seconds when James Maddison slotted past Alex McCarthy before Heung-Min Son and Dejan Kulusevski plundered further goals inside 14 minutes.
Martin responded by hooking forward Kamaldeen Sulemana, making his first of the season, and replacing him with defender Nathan Wood.
Pape Sarr hit his first of the season and Maddison squeezed in a second on the brink of halftime, prompting more boos from the home supporters.
Asked about ripping up his game plan, Martin said: "I think the game is encapsulated by a moment where we have a chance to foul Djed Spence, he shouldn't get that side of us.
Tottenham lead within 40 seconds against Southampton 🤯
James Maddison tucks the ball away calmly 👏
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"We came up with a game plan together as a group of players and staff that everyone really agreed is the best way to be aggressive against Spurs.
"It (was) to try and steal the ball and stop them getting in the flow with their running and to press high.
"We didn't do it with anywhere near enough physicality or aggression within 30-odd seconds of the game. So then Djed Spence wriggles, we don't foul and they score.
"Flynn Downes wriggles away from Madison in the second half and he fouls him. I think moments like that are so big and we don't manage them well enough.
"When we have a setback, it's been the story of the season so far, we don't deal with it anywhere near well enough.
"Take a breath and get back to the plan, stick together, ride out the tough moment and it's game over after 14 minutes."
Kamaldeen Sulemana made his first start of the season against Spurs.Kamaldeen Sulemana made his first start of the season against Spurs. (Image: Stuart Martin) Ghana international Kamaldeen cut a frustrated figure in the dugout after being hooked 15 minutes into the match, arguing with assistant Matt Gill.
"I feel for Kamaldeen but we're 3-0 down in 14 minutes. (You) could take anyone off. But at that point, we needed a defender to plug the gaps," Martin said.
"I knew Kamaldeen was not going to last more than 50 to 55 minutes because he hasn't had many minutes anyway.
"So it was the obvious choice for me. It's not about his performance and I feel bad for him that it's him but we had to do something."