"It's amazing the number of games Gareth has played and even today he proved to be a perfect player for the team," Koeman told the club's website (www.evertonfc.com).
"Scoring the goal to make it 1-1 was an important moment in the game and it was a special day for Gareth but even more special that he scored and the team won, and that was still the most important for today (Saturday)."
The 35-year-old Barry, who has also played for Aston Villa and Manchester City, won 53 caps for England.
"He's very crucial. He's not a big talker in the dressing room but his quality on the pitch, the cleverness on the pitch is important," Koeman added.
"It's also important for Gareth that if the team is playing very compact he doesn't need to run a lot of distance and he is still one of the best midfield players."
Saturday's win gave Koeman's side the best five-game start to a Premier League season with 13 points.
(Reporting by Ian Rodricks in Bengaluru; editing by Sudipto Ganguly)
