At the Rajamangala Stadium, Harhys Stewart gave the Lions the lead going into half-time, before a Kakana Khamyok penalty in the second half levelled the scores on the night.
Minutes later, a Pansa Hemviboon own goal again gave Singapore the lead, but Thailand held on until the final whistle.
Lee said he was “massively proud” of his team’s performance, and that he felt it warranted the game going into extra time.
“Football is cruel, over the two legs maybe we just fell short a little bit,” he added.
Despite the exit, the Lions gave their supporters reason to cheer as Tuesday’s 2-1 victory is Singapore’s first win over Thailand since 2012.
“As a group, we’re very clear with what we’re going to do in terms of building towards the Asian Cup, and I’m sure there will be bumps along the road … as we try to put ourselves against even stronger opposition,” said Lee, who led the team last year to a historic qualification to the Asian Cup.
“During those rough days that will come, I believe that this group of fans that have been backing us will continue to back us because that will be important for the boys.
“They should know that we will never give up, we never give in, we just keep trying and trying and trying as we move towards our target.”