After goalkeeper Gianluigi Donnarumma kept Milan on level terms with two outstanding saves, Franck Kessie had put the visitors ahead in the 78th minute only for Joaquin Correa to rescue a point for dominant Lazio in the 94th minute.
At first, he stayed in his area before finally being coaxed to depart after a few minutes of protesting.
"I didn't understand the reason for the sending off, then he (referee Luca Banti) explained that I had left the technical area," said Inzaghi.
"I apologised, I was so keen on this game and my boys were there.
Donnarumma made a superb save in the first half to deny Ciro Immobile from close range and surpassed himself by managing to claw Lazio defender Wallace's downward header off the line in the 71st minute.
Milan, without suspended striker Gonzalo Higuain and also missing half a dozen regulars through injury, still managed to take the lead when Davide Calabria pulled the ball back and Kessie's shot was deflected into the net by Wallace.
Milan held on until deep into added time when Cristian Zapata headed a clearance straight at Correa and the Argentine controlled the ball with his thigh before scoring with a low shot on the turn.
"It wasn't easy to play with so many players in a role that was new for them," he said. "We are in an emergency but it is no good crying about it." (Writing by Brian Homewood Editing by Ian Chadband)
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