Warner's heroics helped to minimise the impact of batting collapses in both Tests but Smith admitted it was an issue the Australians desperately needed to address.
"We would have much preferred 2-0 but it's nice to get over the line here and have a 1-1 draw," Smith said.
"I still think we have a lot of improvement in us. We probably let ourselves down at times throughout this Test match.
"I thought our first-innings bowling was very good to restrict them. Our first-innings batting, obviously the partnership between (Peter Handscomb) and Davey was fantastic.
"And then we got ourselves into one of our collapses that we've had.
"I think we've had 15 collapses in our last 14 games ... that's not good enough for an Australian cricket team.
"That's something we really need to work on. We need to rectify that come the next series and the Ashes."
Offspinner Lyon proved the hero for Australia in Chittagong, backing up his seven-wicket haul in Bangladesh's first dig to record career-best match figures of 13-154.
Australia were forced to endure some nervous moments with Warner and Smith departing cheaply before Glenn Maxwell scored the winning runs with a six over midwicket.
A 2-0 series loss would have sent Australia tumbling to No.6 on the world Test rankings, their lowest point in almost 30 years.
Lyon's 6-60 in the second innings took his wicket tally for the tour to 22, just one shy of Sri Lankan tweaker Rangana Herath's record for the most dismissals in a two-match Test series.
An innings victory for Australia had looked like a genuine possibility when Bangladesh collapsed to 5-43 before lunch.
But Mushfiqur and Mominul Haque (29) led a spirited rearguard action to keep the hosts in with a chance.
Pat Cummins (2-27) was forced to battle through stifling heat but still managed to rattle the Bangladeshi batsmen with sheer pace and an array of short-pitched deliveries.
Matthew Wade made an important contribution with the gloves, stumping danger man Tamim Iqbal (12) and Shabbir Rahman (24) both off the bowling of Lyon.
