The result ended Lazio's four-match winning run and left Igor Tudor's side seventh, three points behind city rivals Roma - who occupy fifth place and the final Champions League qualifying position.
Sixth-placed Atalanta are only a point ahead of Lazio, but with two games in hand.
At the other end of the table, relegation-threatened Sassuolo recorded a surprise, but crucial, 1-0 victory over newly crowned Serie A champion Inter Milan.
Things appeared to be going to plan for Lazio when they took the lead in the 11th minute.
Monza goalkeeper Michele Di Gregorio did brilliantly to palm Daichi Kamada's deflected strike onto the bar, but Ciro Immobile tapped in the rebound.
It was the Lazio captain's first goal since February.
Monza went close several times and equalised in the 73rd minute when Duric fired in the rebound after a great save by Christos Mandas to parry Giulio Donati's header.
The goal was initially ruled out for offside but that decision was overturned by VAR.
Donati was the villain 10 minutes later, though, as his woeful back pass was intercepted by Matias Vecino, who fired past a stranded Di Gregorio and into the empty net to restore Lazio's lead.
But Duric headed in Matteo Pessina's cross in the second minute of stoppage time to earn his side a share of the spoils.
Inter, who won their 20th Serie A title last month, suffered only their second league defeat of the season - their first, way back in September, was also against Sassuolo.
Armand Lauriente returned from suspension and his 20th-minute strike was enough to give the hosts only their second win in their last 16 matches.
Lautaro Martinez thought he had equalised for Inter on the stroke of halftime but his effort was ruled out by VAR for offside.
Sassuolo moved to within two points of safety, just below Hellas Verona, Frosinone and Empoli - who are all on the same number of points.
Empoli host Frosinone on Sunday, while Verona welcome Fiorentina.
