Saints stunned the Premier League leaders 2-1 at St Mary's, with Shea Charles hitting an 85th minute winner.
Afterwards, a delighted Eckert said: "I'm young, it's not that I've been doing this for a long, long time, so definitely the biggest game so far. It was a big night for us as a football club.
"For the supporters, the stadium has been buzzing from the first minute, and the reality is championship football from tomorrow onwards.
"I think the game always lives off moments. We had some very good ones at the start of the game, even to open up the scoreline a little bit earlier.
"I think we deserved them to go up, and then you know the second half unfolds, and you have some substitutions that you need to deal with.
"They can change the game in both directions. We were prepared to be able to react to that, and the boys pushed through the 90 minutes very well."
On his emotions at the final whistle, Eckert added: "We said this at the beginning of the week, that we have put ourselves in a position to play for something in April and May.
"Then it's full preparation to go into a game all out, and then you need to close that chapter quite quickly, because that's just the reality.
"From tomorrow onwards, it's Championship football. They can enjoy it tonight. Whoever needs to do regeneration comes in to recover tomorrow.
"But the reality is that we have a big game coming up on Tuesday. That's the balance, and the test of maturity that we have from tomorrow onwards."