Martin, who guided Saints back to the Premier League last season, was sacked on Sunday evening following a 5-0 home defeat to Spurs.
He departed St Mary's with Saints bottom of the Premier League table with just five points and a -25 goal difference from 16 games.
Interim boss Simon Rusk took Wednesday's 2-1 Carabao Cup quarter-final defeat to Liverpool, in which Archer scored his fifth goal of the season.
Southampton sacked Russell Martin after a 5-0 defeat to Spurs.Southampton sacked Russell Martin after a 5-0 defeat to Spurs. (Image: PA).Archer previously explained to us that he joined the club after Martin convinced him a move to the South Coast was the correct step in his fledgling career.
Asked about Martin's departure, Archer told the Daily Echo: "It's obviously disappointing but it's football. Things happen in football.
"It's part of the game and you have to move on from it. It's always disappointing when someone loses their job. He was close to us but it's a part of the game, we have to move on."
Over 72 hours after sacking Martin, Southampton's club hierarchy is continuing to search for his permanent successor.
Rusk will prepare the team to face Fulham on Sunday until he is told otherwise. The players have a day off on Thursday before returning to Staplewood.
Sport Republic's Henrik Kraft and Rasmus Ankersen in attendance on Wednesday, alongside CEO Phil Parsons.Sport Republic's Henrik Kraft and Rasmus Ankersen in attendance on Wednesday, alongside CEO Phil Parsons. (Image: Stuart Martin) Asked how much the players know about a managerial appointment, Archer said: "You don't have a clue until they walk into the building. No clue on that one."
Southampton's first action of the post-Martin era was to go long from kick-off, and it yielded them a free-kick in a dangerous area.
A clearance from Nathan Wood was cheered by the Northam Wall in the first-half and Archer was asked to hold up the ball at times.
The different approach did not prevent Liverpool from taking a two-goal lead in the first half through Darwin Nunez and Harvey Elliott.
Saints responded as the handbrake was released in the second half and the hosts increasingly looked to play forward quickly.
Cameron Archer, what a goal! 😍 pic.twitter.com/Z3gq5r9NUT
— Sky Sports Football (@SkyFootball) December 18, 2024
A top save from Caoimhin Kelleher denied Archer a quickfire second when he flicked on an inviting delivery from Yukinari Sugawara.
Southampton nearly profited from an Alex McCarthy long ball in additional time but Jarell Quansah's foul on Mateus Fernandes went unpunished.
Cameron Archer and Nathan Wood celebrate goal against Liverpool.Cameron Archer and Nathan Wood celebrate goal against Liverpool. (Image: Andrew Matthews/PA)."We didn't really have a way of playing," goalscorer Archer explained. "I would say it felt quite freeing.
"We made the most of it. We used our abilities and what we were capable of as players. Our individuality.
"Adam Lallana and the skipper (Jack Stephens) spoke with the gaffer. They spoke about togetherness. It's down to us and what we do on the pitch.
"In the second-half, we got a bit closer to them. We applied more pressure going forward. We had a right good go at them. We dug deep and we stuck to the gameplan quite well."