The Toffees were sub-standard in the first-half and took time to improve after the restart, though Fraser Forster made some good saves to keep out Romelu Lukaku.
The long-awaited breakthrough finally arrived in the 73rd minute when substitute Enner Valencia tapped in the rebound from Forster's save from Lukaku.
And that invited a late flurry as Leighton Baines scored from the penalty spot following a foul on Valencia and Lukaku fired high beyond the goalkeeper in the closing minutes for a much-needed win to open 2017.
Yet the Dutchman, speaking in his post-match press conference, refused to take credit for the impact of Valencia after he came off the bench.
Koeman: Too early to decide on Valencia's future
When asked whether he thought it was his decision to bring on Valencia that changed the game, he said: "No. I think at that stage, we needed something different up front. That was the reason to start Dominic [Calvert-Lewin] behind Romelu.
"Because I wasn't happy with the support to Romelu from the midfield in the last few weeks. That was the reason to start Dominic.
"We needed to make decision to change the boy. You bring Enner straight in or you wait also for the last part of the game.
"That's what happened after one hour. Okay, he was one of the players who changed the game, who scored the first goal, and created those difficulties to the defenders of Southampton."
On whether he was surprised by West Ham United allowing Valencia to go out on loan to Goodison Park, Koeman continued: "We had the possibility to get the boy on loan here at Everton and it's good because he can play in different positions attacking-wise.
"He always gives to the manager a lot of possibilities to change systems, to have the boy in front. The final decision what we need to make on his future, it's still too early for that."
Toffees boss says last 30 minutes "made the difference"
Koeman was also pleased with the late performance of Lukaku, his hold-up play allowing the home side to keep the pressure on in the last quarter of an hour.
He added: "Romelu is always that target man, that strong boy in front, what you need to play football.
"When we're pressed, we have the option with Romelu. We did that much better in second-half than in the first but overall we didn't have that intensity from the start.
"We spoke about it. Maybe that's the reason that in the last 30 minutes, Everton were much better than Southampton, because they had one less day to recover from the last game.
"We made the difference in the last 30 minutes."
Baines had to be reminded of his penalty-taking duties late on, with Lukaku appearing keen to take over - Koeman commenting: "Everybody likes to help Romelu to be top scorer in the Premier League.
"But it's still 1-0 and Leighton is the first on the list. If it's 2-0 in the last minute of the game, then I can understand Lukaku taking it. But it was a very important moment to kill the game at that time."
'We need to wait for Calvert-Lewin diagnosis'
Koeman also explained that the club would need time to assess summer signing Calvert-Lewin, who was forced off with an apparent ankle injury just 12 minutes into his first Everton start.
He continued: "We need to wait for tomorrow [to discover the extent]. I don't know exactly what happened. I'm disappointed, but he's young - he will come back."