The Foxes fought back from two goals down to win as Facundo Buonanotte, Jamie Vardy (penalty) and Jordan Ayew struck for the visitors. Ryan Fraser was also sent off for the Saints, which had a two-goal lead through Cameron Archer and Joe Aribo.
Martin later said: "I think the sending off changes the game. We started the game so well, the second half so well, their goal came out of nothing. Then Ryan gets sent off and the game becomes really difficult.
"I thought we were better in the start of the second half then the end of the first half. We didn't take our chances to get the third goal. Then someone gets sent off and it changes the whole context.
"I think with VAR I can't have any complaints, it is what it is.
"It's the referee's prerogative to add the time on he feels has been added on in injury time. He told our players it was the final action, so we have to defend it properly."
On the message to the players, he continued: "To understand what's hurt us, to understand what has put us into a 2-0 lead in a really brilliant performance. But right now there is no message I can give that is helpful because of the emotion everyone is feeling. Give everyone to digest that and the process will continue.
"The fans have been brilliant since I've been here. I'm not sure the boos were directed at their team because the team were giving everything. My message is to be really supportive, they've been great."
On the message to fans questioning his future, Martin added: "I've nothing to say apart from that I need to keep working and we need to win a game of football. I hope they will always turn up and support their team, they've been amazing."