Marco Parolo had snatched the lead for the visitors just before the break before three goals in five second-half minutes settled an enthralling encounter.
The striker scored in the first half before Rodrigo Palacio equalized just before halftime, only for Marco Parolo to restore the visitors' lead a minute later.
Palacio and Fredy Guarin turned the game on its head early in the second period, but Sansone added his second - and the third goal in five minutes - to ensure the spoils were shared.
Having accused his side of delivering its worst performance of the season in a 1-1 draw at home to Sampdoria last time out, Inter coach Walter Mazzarri made two alterations, with Andrea Ranocchia and Mateo Kovacic coming in for Rolando and Saphir Taider.
Parma made only one change to the team that drew with Bologna eight days before as former Inter man Walter Gargano came in for Afriyie Acquah, but they swapped over again in the first quarter of the match as the injured Uruguayan was forced off.
The visitors impressed early on and after Gabriel Paletta squandered a chance, Sansone made no mistake in the 11th minute.
Marco Marchionni fed Sansone and the forward confidently slotted the ball into the bottom-left corner for his first goal in Serie A this season.
Palacio and Sansone should each have scored for their respective sides as the first half went on, but the Argentine did eventually bring Inter level in the 44th minute.
Jonathan fired a fierce cross into the six-yard box and the attacker poked home, but the hosts were behind again before halftime as Handanovic dropped a Paletta shot to leave Parolo the simple task of knocking into an empty net.
The opening stages of the second half were rather low-key, but the game soon exploded into life again as Inter scored twice in three minutes.
Palacio headed in Ricky Alvarez's 54th minute cross, while Guarin saw a left-footed half-volley from the edge of the area deflect past Antonio Mirante.
Donadoni's men were soon back on level terms again as Mattia Cassani produced a fine low cross from the right and Sansone held off Esteban Cambiasso to score his second.
After a brief lull in proceedings, Palacio went close to completing his hat-trick in the 72nd minute as he headed Yuto Nagatomo's left-wing cross at Mirante, and Kovacic drilled just wide from long range.
Parma optimistically asked about a penalty decision in the 86th minute as Sansone's shot smashed against Hugo Campagnaro's arm amid a spell of pressure, but Inter held on and almost snatched an injury-time winner when Cambiasso shot straight at Mirante.
Inter remains fourth in the table following the draw, while Parma moves into 10th, level on points with Cagliari.