The first-half debate between captain and coach summed up the challenge for France who have found the back of the net only twice in three games at the tournament: once from an own goal against Austria and a penalty scored by Mbappe against Poland.
The draw - which means France finished second in Group D, the first time in Deschamps' 12 years in charge that they have not topped a group stage pool - will do little to settle the unease among fans at their team's lack of goals.
As in previous matches in Germany, France looked dangerous with wingers Ousmane Dembele and Bradley Barcola - making his first start in the national team as Deschamps reworked his attack - offering lots of threat.
Mbappe, back from sitting out the 0-0 draw with the Netherlands, seemed uninhibited by the mask he wore to protect the broken nose that he suffered in the 1-0 opening win over Austria, powering his way past defenders with ease.
The Real Madrid-bound striker showed signs of working well with Barcola before breaking the deadlock to score his 48th goal in 81 appearances for France, but his first at a European Championship, from the penalty spot in the 56th minute.
In a sign of how France's dominance, their 55 touches in their opponents' penalty box were more than any team in a match so far in Germany, according to Opta.
Deschamps turned to experienced forwards Antoine Griezmann, who he had dropped to the bench, and Olivier Giroud in an attempt to put the match beyond doubt.
But it was Poland's veteran captain Robert Lewandowski who scored the game's only other goal, also scoring from the spot with a re-taken penalty.
Deschamps was left to express his frustration once again at France's lack of ruthlessness. "We have to accept it, even if we did what we did by creating the chances," he said.
"At least we’re qualified and we know when we’ll play. A new competition will start. We’re not going to live with regrets. We could have been more efficient, we were not able to double our advantage."