Denis Bouanga put the MLS club ahead in the 84th minute but substitute Wallace Yan responded two minutes later to ensure Flamengo would progress unbeaten to their last-16 tie against Bayern Munich in Miami on Sunday.
With top spot in Group D already assured, Flamengo coach Filipe Luis made seven changes to his starting side but there was no change to the style that already earned the Rio de Janeiro club wins over Chelsea and Esperance.
With a little more accuracy in front of goal, they would have comfortably beaten an LAFC side who were already condemned to a group stage exit after losing their first two games without scoring a goal.
"We had a great group stage," Felipe Luis told reporters.
"We achieved our objective early so we've been able to bench players with yellow cards and give minutes to players that hadn't played yet.
"We had rhythm and we had chances to score. Unfortunately, it didn't happen. We are very happy, no injuries and a great group stage, and now we're thinking about Bayern and nothing but Bayern."
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