The Russians failed to reach the last 16 of the competition as United earned a narrow 1-0 win at Old Trafford.
Costa believes that Shakhtar's failure to convert their opportunities in the first half had an effect on the overall tie.
"Our failure to take our chances [in the first half] had a real impact on the game because we could have scored those goals but we lost - that's football," Costa told UEFA's official website. "We are learning and now we have to turn the page and try again next season."
Bayer Leverkusen will join United in the next round thanks to their 1-0 victory at Real Sociedad.