Hellas looked to have secured all three points when substitute Romulo gave them a 3-2 lead with seven minutes remaining.
Lazio, who had Senad Lulic sent off for a straight red soon after, drew level when Mauri tapped in his own missed penalty.
Keita had scored the first goal of the game for Lazio on the half-hour mark but Hellas striker Marquinho equalised seven minutes later.
Lulic put the home side back in front after one hour of play only for Juan Manuel Iturbe to make it 2-2 nine minutes later.
The outcome at the Stadio Olimpico has left both sides in the top half of the standings, level on 53 points.
They are two points adrift of Torino, who are sixth and hold the last Europa League qualifying spot with two games remaining.
The hosts had an opportunity to score early. Antonio Candreva's 20-yard shot forced a good dave from Hellas goalkeeper Rafael.
Hellas should have gone in front in the 12th minute when Emil Halfredsson set up Luca Toni in the area but the veteran forward shot wide.
The hosts went ahead when a fast counter-attack saw Keita get behind Candreva's cross and his poweful 15-yard shot beat Rafael.
Lazio's joy only lasted a few minutes.
Hellas pulled level when Marquinho drilled a right-footed effort towards the far post and out of Lazio goalkeeper Etrit Berisha's reach.
The home side missed a good opportunity to go back in front early in the second half as Mauri's powerful header from Candreva's cross from the right went wide of the near post.
Lazio got their second goal when Keita's pass found Lulic in the box and he drilled a right-footed strike past Rafael.
Hellas restored parity when Jacopo Sala served Iturbe inside the area and his left-footed effort went past Berisha.
The visitors continued to press forward and went in front when Romulo struck from close range after a neat cross from Hallfredsson.
The Roman club lost Lulic to a red card after a hard foul on Iturbe but were still next to find the net.
The referee pointed to the spot after Michelangelo Albertazzi brought down substitute Miroslav Klose in the area, with Albertazzi also seeing red.
Although Rafael guessed correctly the direction of Mauri's spot-kick, he spilled the ball and Mauri pounced to score the equaliser.