Everton returned to winning ways in the English Premier League on Saturday by defeating West Brom 3-0 at Goodison Park to maintain their seventh place position in the table.
Koeman was delighted with his team's "professional" performance and was particularly pleased with the way they dealt with West Brom's aerial threat.
Everton took the lead in 39th minute when Kevin Mirallas pounced to tap in after Ben Foster pushed Barkley's shot straight out into the area and Schneiderlin made it two before Romelu Lukaku headed home Barkley's ideal second half cross.
Tony Pulis's side offered little in the second half and the home team capitalised with Lukaku scoring with a header, his tenth goal in his last ten games.
Barkley has five assists in his last eight games - only Swansea's Gylfi Sigurdsson has more in 2017 - and beginning to impose himself on matches more regularly.
"We will always score because we have those types of players".
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"Arsenal are normally strong and have good players, so it will be really tough to catch them".
"Maybe I need to sign that contract, so we can finish with the stories!"
At the other end the weak link in the Baggies' defence had looked like being Allan Nyom and it was in the left-back's vicinity where the move for the hosts' opener emerged.
"It was a well-deserved win and we were very professional", Koeman added.
'We will be solid enough to be competitive in the Premier League every year, and have good cup runs. "I think of course he will be in the squad".
"Of course, you can't compare the two teams - but we played well today, controlled the game and we were risky in certain moments of the game". "He has a lot more freedom now in the team".