Brabantio learns of Roderigo's name but not of Iago's. Iago also manages to manipulate Othello, when talking of killing Cassio. Iago adds into the conversation "But let her live". This sounds like Iago is trying to save Desdemona, but more likely is that Iago is deliberately planting the idea of killing Desdemona into Othello's mind. Othello replies with "Damn her, lewd minx. O damn her, damn her". From that reaction we can tell that Iago has reminded Othello of what she has done, and rather than agree with what he has said it sounds like he is more determined to kill her. So Iago has succeeded in condemning Desdemona to death at Othello's hands.
Iago also has a nastily good way of letting Othello's imagination do all the work convincing Othello that Desdemona has been unfaithful. He does this most effectively in his vivid description of Cassio dreaming of having sex with Desdemona. Finally Othello says to Iago "Give me a living reason she's disloyal", he is asking for evidence that Desdemona and Cassio have been sleeping together. Iago now conjures up a story of Cassio dreaming of Desdemona. "In sleep I heard him say. "Sweet Desdemona let us be wary, let us hide our loves". And then sir he would gripe and wring my hand, Cry "O Sweet Creature!" and then kiss me hard, As if he plucked up kisses by roots That grew upon my lips; then laid his leg Over my thigh, and sighed, and kissed, and then Cried "O cursed fate that gave thee to the Moor". This flamboyant description would have run havoc with Othello's imagination, being forced by Iago to think of his wife sleeping with another man. The fact that Iago also lies that Cassio has warned Desdemona not to let Othello find out would only go further to infuriate Othello. Luckily for Iago this seems to be all the evidence that Othello needs to make the judgement that Desdemona has been unfaithful.
Iago's effectiveness at noticing innocent situations and making them look suspect is another way in which he gets Othello to believe of Desdemona's unfaithfulness.