Lyon surged into a 2-0 lead at the Etihad Stadium thanks to first-half goals from Maxwell Cornet and Nabil Fekir.
Bernardo Silva halved the French outfit's advantage mid way through the second-half. But Bruno Genesio's men held on for a famous victory.
Arteta, who took charge of the City squad due to Pep Guardiola's suspension for being sent-off in last season's quarter-final exit to Liverpool, said: "When you are not at the races at the start of a game in the Champions League, it doesn't matter how good you are afterwards because it is too late."
Both of Lyon's goals came from careless passes in dangerous positions. The sloppiness helped Lyon play better than their form had suggested.
While City have collected 13 from a possible 15 points in the English Premier League, Lyon have only seven points from their first five fixtures in the French top flight.
"We need to be capable of repeating this performance in order to fulfil the objectives we have," said Genesio. "The end of the game was tougher as they brought it back to 2-1 and brought on talented players."
Genesio added: "We would have taken 2-2 before the match but given the way the game went we'd have been disappointed not to leave with the three points."