In a remarkable climax to the Ligue 2 season, the top six were separated by three points before Friday's games.
Amiens were set to finish sixth - and not even make the play-offs - when Emmanuel Bourgaud scored a 96th-minute winner at Reims to trigger a pitch invasion from club officials and players.
Club president Bernard Joannin said: "It is an extraordinary moment.
"I saw a fighting team again. At the start of the season, our ambition was to stay up - and now we are in Ligue 1.
"Football is a play every Friday night. We gave this evening an extraordinary drama."
Joannin also said that he had offered new French president Emmanuel Macron - born in Amiens - a signed shirt before the game which had brought them both luck.
Strasbourg sealed the title after a 2-1 home win over Bourg en Bresse.
Third-placed Troyes came from two goals down to beat Sochaux 3-2 but Bourgaud's late goal for Amiens means they now face a play-off against the side who finish 18th in Ligue 1 following the final round of top-flight games on Saturday.
Both Strasbourg and Amiens have now won back-to-back promotions, with the latter having never played top-flight football during their 116-year history.
It is also a fourth promotion in six years for Strasbourg, who return to Ligue 1 for the first time since 2008, having entered liquidation in 2011 and been placed in the fifth tier.
Final-round results
Strasbourg 2-1 Bourg-en-Bresse
Reims 1-2 Amiens
Sochaux 2-3 Troyes
Lens 3-1 Niort
Brest 6-2 Ajaccio
Laval 1-2 Nimes
Amiens were promoted to the top flight of French football for the first time with a stoppage-time goal that carried them from sixth to second.