The Serbian striker finally found form after failing to strike through his first four games for the club, opening the scoring just before half-time by turning home Antonio Candreva's low cross.
Djordjevic then curled home a second in the 75th minute and completed his treble eight minutes later before Marco Parolo netted in the second minute of injury time to cap a superb second-half showing from the visitors.
But, while Lazio avoided a third straight defeat, there was more misery for struggling Parma with a 4-2 defeat at Udinese.
Roberto Donadoni's men made a great start to the contest at the Stadio Friuli, midfielder Jose Mauri giving Parma the lead with a cool finish after 22 minutes.
A double from Udinese talisman Antonio Di Natale had seemingly turned the game around, only for Antonio Cassano to restore parity with an excellent chipped penalty after Silvan Widmer had fouled Mauri.
But the visitors could not carry the momentum from that goal into the second half and fell behind again in the 58th minute, Thomas Heurtaux's magnificent overhead kick restoring the hosts' advantage.
Parma were then reduced to 10 men as midfielder Afriyie Acquah saw red for a second bookable offence following his foul on Luis Muriel.
Udinese duly capitalised on their numerical advantage, Cyril Thereau making the points safe six minutes from time to seal a win that moves Andrea Stramaccioni's side back to third in the table.
