The win, only Everton's second of the season, pulled the 'Toffees' out of the bottom three, though only a late turnaround prevented Koeman's side from suffering their fifth straight League defeat without finding the net.
The Dutchman was indebted to striker Oumar Niasse, who scored twice after coming on as a substitute to propel his side to three points.
Niasse was infamously forced to train with the club's U23 side just a year ago as Koeman looked to force him out of the club but as we know, things change quickly in football.
Koeman alluded to that very fact in his post-match press conference, though he described speculation that he was under pressure going into the game as 'ridiculous'.
'For me, the players, for everyone it (the result) is important. Everyone knew it was a must-win game,' said Koeman.
'Sometimes we go too fast - if you win you are sure about your job and if you lose you are not sure about your job, but that is ridiculous.'
Whilst it is crazy for a manager to be under pressure so early on in the season, it is unlikely that Everton's owners will stay patient forever, especially given that Koeman spent over £150m in the summer transfer window.