A lightning start to the match saw the two teams equally share eight goals and trade the lead in the opening four minutes and 12 seconds, before Albin Lagergren and Hampus Wanne opened the first two-goal lead of the match, putting Sweden 6-4 up.
France were soon back on level terms, before strikes from Dylan Nahi, Melvyn Richardson and Theo Monar turned 9-9 into a 12-9 advantage for the visitors in the 19th minute.
But Michael Apelgren's men were not about to let the group leaders pull clear - Feliz Moller's goal five minutes before the break drew them level again at 13-all, before five French strikes were matched by the hosts to end the half at 18-18, the final equaliser scored by Moller with three seconds to spare.
Early in the second half, it was Sweden's turn to keep edging in front, but once again they seemed unable to carve out a lead of more than two goals.
France, who had won all four of their matches in the group before tonight, regrouped and soon found themselves back in front, and they would have had a healthier lead had Andreas Palicka not pulled off several top class saves to keep Sweden in it.
Lukas Sandell took the roof off the Scandinavium with five minutes to go, when his shot found the top corner to level the scores at 31-31, but the hosts could not kick on, and two goals from Richardson looked to have sealed the win.
In the final minute, Felix Claar made it a one-goal game, and a France miss gave Sweden the chance to find an equaliser, but Sandell's shot with three seconds to go was blocked, meaning a 33-32 win for "Les Experts".
Both teams shared the goals around, with France's Nicolas Tournat the match's top scorer with six goals, while Sebastian Karlsson was Sweden's most prolific, with five.
Sweden are on four points after five matches in the group, remaining third behind Denmark, and they finish their EHF Euro Cup campaign away to the Danes on Sunday.