Inter have only won one of their past six league games at San Siro after coming from two goals down to claim a point in Milan.
In-form Brazilian striker Felipe Anderson put Lazio ahead after just two minutes and produced another clinical finish to double the advantage before half-time.
However, Inter rallied after the break and Kovacic fired in a stunning volley from the edge of the area before Rodrigo Palacio equalised 10 minutes from the end.
"It's my best goal but we are disappointed because we could have won," Croatia international midfielder Kovacic told the club's website.
"We are not happy, because we made mistakes to allow them to score their goals.
"We did well to recover, but when you go two goals down it becomes difficult to win."
Kovacic accepts that 11th-placed Inter are struggling to put a series of good results together this season, but insists they can climb up the table after Christmas.
"We lack continuity but we are working with the coach and I'm sure we will improve," added the 20-year-old."
