Mario Balotelli struck from the spot in the 85th minute, but only after a discussion over whether he, stand-in captain Jordan Henderson or striker Daniel Sturridge would be taking the penalty.
Injured captain Steven Gerrard said on ITV Balotelli had disrespected Henderson - who was due to take the penalty.
But Rodgers refused to be drawn on the incident, instead wanting to focus on his team's hard-fought victory.
"We got the win, which was the most important factor tonight," the Northern Irishman told a media conference.
"Steven should have taken it as he is our penalty taker. I have a number of outstanding penalty takers in the squad with Jordan, Mario, Rickie Lambert and others.
"You saw the spirit of the team tonight. To get the victory was excellent."
Rodgers was asked to clarify if Henderson was down to take the penalty, Rodgers said: "I’m clarifying that we won the game 1-0."
The 42-year-old lauded his team for their performance, and was particularly pleased with Simon Mignolet's clean sheet.
"It was an excellent result and performance. It was always going to be a tight game," Rodgers said.
"Besiktas came here with big confidence and I believe they have scored in every away game they have played this season.
"So it shows you the level of our defensive organisation that we kept a clean sheet. Simon had a really good save in the first half, an outstanding save it was.
"On top of that, we created chances. We worked the ball much quicker in the second half. It sets us up perfectly, a great result for the second leg - a win and a clean sheet."
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