In Othello Act II, scene iii, Iago uses wine and Roderigo to bait Cassio into fighting. In the end, the drunk Cassio wounds Montano, governor of the island, and loses his rank of lieutenant; Roderigo can't be found; and Iago comes out blameless.
I suppose it doesn't matter who starts the fight, only that Iago is behind it. When Othello is waked from his honeymoon night with his new bride, he doesn't care who starts it: he doesn't go looking for Roderigo. Othello only cares about the following: Montano is wounded; Cassio is drunk; and he (Othello) is both disturbed and made to look like he can't control his men.
The consequences of the fight set up the action for both Cassio's and Desdemona's demise. First, Cassio is demoted. Next, Desdemona will ask Othello to forgive Cassio, which only makes her look like she's romantically involved with him. All of these will lead to the rifts between Othello and Cassio and between Othello and Desdemona. Both of these rifts, ironically, bring Othello closer to Iago, the villain.