Despite early setbacks with Ronald Araujo lost to injury and Betis taking the lead, Barcelona ended up being the winning side.
"The results have helped us to believe and the team has used their mentality to win this game," Koeman said as he assessed the game.
"This is the way."
The hero of the hour was Francisco Trincao, who came on as a substitute and scored with just minutes left on the clock.
"He's young, he has to adapt to the level of a big club like Barcelona," Koeman explained.
"We've been giving him minutes.
"In the last few games, he's had bad luck missing chances.
"Today he was decisive and that's important for him and for us."
Another impactful substitute was Lionel Messi.
"Barcelona with Messi is a better team, much more effective," Koeman admitted.
"He's been here many years showing that he's a vital player for this team."
Some were surprised that Messi and others were on the bench, but Koeman underlined his rotation policy.
"We've made rotations for players with many minutes and games, although Araujo's injury has made it even more complicated," he explained.
"You have to highlight the team mentality, in a hard game we've won with our quality and character."
The coach was also asked why he preferred to bring on Frenkie de Jong for Araujo, despite having Sergino Dest and Samuel Umtiti on the bench.
"I like to have a right-footer and a left-footer as a pair of central defenders and Dest had discomfort yesterday and wasn't ready to play so long, so we went for Frenkie," he said.