However, they now face a run of four successive matches at Goodison Park and Koeman admits they need to take advantage to dampen talks of a crisis, with the club languishing 18th in the Premier League.
"I am always enjoying football", smiled the Dutchman.
'You can't compare Valencia with Everton or Southampton or Benfica or PSV because that was really hard, ' said Koeman. Add to that the fact that Everton failed to test Begovic regularly, and one sees the same problems from every match this season.
As Koeman struggles to accommodate all his new signings after summer spending headlined by the return of Wayne Rooney and addition of record signing Gylfi Sigurdsson, it is a rather compelling and refreshing paradox that two young players costing a combined £1.5 million are the ones adding excitement and attacking objective in recent matches.
The midweek victory saw impressive displays fromÂ Dominic Calvert-Lewin and Oumar Niasse in particular, with both finding themselves on the scoresheet and ending Everton's drought.
The biggest criticism of Ronald Koeman this season is his unflinching desire to fit all of his new signings in at once, which at times has led to three attacking-midfielders on the pitch at the same time, getting in each other's way and not offering much going forward.
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He has also promised to address some of New Zealand's most pressing social issues such as child poverty and a housing shortage. That will likely have implications for the New Zealand dollar, the world's 11th most-traded currency in 2016.
Now, the 36-year-old can lend his knowledge and game management skills to Tony Pulis's squad as they look to continue bridging the gap between themselves and the Premier League's top six.
"All the credit to Oumar Niasse, that is all about life".
Barry, 36, joined Tony Pulis' side just before the start of the new season and has quickly established himself as a regular starter for West Brom, who are preparing to take on Arsenal at the Emirates Stadium on Monday (25 September).
"I thought we were the dominant team", Howe told BBC Sport.
Despite spending most of the first 45 defending, it was Bournemouth who drew first blood three minutes after the restart. If you play for titles, or for European football, the next game is always the most important.
"For the first goal, he won the ball in midfield, made the one-two and it was a great goal".
Fair play to him for working his ass off, and taking that chance when it came.