Inter boss Roberto Mancini glowered from the touchline after watching Felipe Anderson score his first brace in Serie A – the Lazio midfielder’s second goal arrived at the end of a wonderful run – but the hosts’ luck would change.
Kovacic halved the deficit with a stunning volley, charging the Inter team with confidence and putting Rodrigo Palacio in a position to score his first goal of the season, a goal that spared his side from defeat.
The point was enough to lift Lazio into third place heading into the winter break while Inter remain in 11th.
Earlier on Sunday, Sampdoria missed the chance to move into the final Champions League spot after they were forced to rescue a 2-2 draw at home to 10-man Udinese.
The Genoa club would have leapfrogged Napoli with victory at the Stadio Luigi Ferraris but, despite the fact Pedro Obiang boosted them into an early lead, goals from Alex Geijo and Danilo gave Udinese the upper hand.
Manolo Gabbiadini would peg the ninth-placed team back with a goal on the hour mark yet they could not quite take advantage of Thomas Heurteaux’s dismissal and had to settle for a single point.
Atalanta fought back from a two-goal deficit to earn a 3-3 draw against Palermo in Bergamo.
Luca Rigoni put Palermo ahead in the sixth minute and Franco Vazquez doubled the advantage shortly after.
German Denis earned and converted a penalty to reduce the arrears in the 40th minute before Vazquez scored his second of the day to give Palermo a 3-1 lead at half-time.
However, second-half goals from Maximiliano Moralez and Denis earned the home side a deserved point.
Kamil Glik scored twice in the second half as Torino came from behind to beat Genoa 2-1 for their first league victory in nearly two months.
Iago Falque gave Genoa the half-time lead but Glik equalised seven minutes after the restart. The Polish defender then headed home the winner in the 63rd minute to condemn Genoa to their first defeat on the road this season.
In a battle between two Tuscan teams in Florence, Empoli held Fiorentina to a 1-1 draw to extend their unbeaten run to five games.
Peruvian midfielder Juan Manuel Vargas scored his first goal of the season to put Fiorentina ahead in the 44th minute but Lorenzo Tonelli equalised shortly before the hour mark.
Elsewhere, Alberto Paloschi scored the only goal of the game to hand Chievo a narrow win at Hellas Verona in the city derby.
The Italian striker headed home Ervin Zukanovic’s cross nine minutes from time as Chievo won for the fourth time this season.