Maignan started things when he smothered a poor effort from Moise Kean.
Then in first-half injury time, De Gea leaped to his right to push away a spot kick from Theo Hernandez.
After the break, De Gea was at it again, this time denying Tammy Abraham.
Yacine Adli, who is on loan from Milan, put Fiorentina ahead in the 35th. Christian Pulisic equalised for Milan in the 60th with an acrobatic volley and then Albert Gudmundsson won it for Fiorentina with a long-range strike to finish off a counterattack.
De Gea, the longtime starter at Manchester United who didn't play anywhere last season, also played a role in Gudmundsson's goal, providing a long ball that Kean knocked down to leave for the Iceland striker.
Juventus conceded their first goal of the Serie A season when Razvan Marin converted a late penalty for Cagliari in a 1-1 draw.
It was a rough ending for Juventus, who had winger Francisco Conceição sent off moments later following a second yellow card.
Dusan Vlahovic converted a penalty for Juventus early on.
Juventus nevertheless remained the only unbeaten team in Serie A after Empoli lost 2-1 at Lazio.
Cagliari almost completed the fightback in the fifth minute of stoppage time but Adam Obert's strike came off the near post.
Juventus are three points behind leaders Napoli, who beat Como 3-1 on Friday.
Lazio missed a penalty but still fought back to become the first team to beat Empoli this season for a fourth straight win in all competitions.
Mattia Zaccagni headed Lazio level in first-half stoppage time to cancel out Sebastiano Esposito's early goal and it seemed as if the home side would turn the match around immediately after the interval when Empoli defender Giuseppe Pezzella tripped Boulaye Dia in the penalty area.
However, Devis Vásquez parried Taty Castellanos' spot kick and the forward nodded the rebound over.
Lazio scored what was to prove the winner six minutes from time when substitute Pedro raced onto Castellanos' through ball and thumped it in off the underside of the crossbar from a tight angle.
Roma were held to a 1-1 draw at Monza in another disappointing result for the team following a loss at Swedish club Elfsborg in the Europa League.
Artem Dovbyk put Roma ahead after the hour mark but Dany Mota equalised for Monza nine minutes later on a play began by new Italy callup Daniel Maldini.
Roma were without Paulo Dybala, who has a muscular problem.
Bologna were held to a 0-0 draw by Parma in an Emilia-Romagna derby despite the visitors playing most of the second half with 10 men after defender Woyo Coulibaly was shown a straight red card for a dangerous tackle.
