There had been suggestions new arrival Thiago Silva would take over the armband following the veteran Brazilian's move from Paris Saint-Germain.
But Lampard has kept the Spanish full-back in the role even if Reece James could play at right-back in the majority of games this season.
He said: "Yes, absolutely Azpi is club captain this season, he's done a great job of it.
"Since I've been in the role, he's been a huge help to me, how's he's handled himself on and off the pitch, how much he feels for the club, how much he communicates within the squad.
"We can all see the quality of Reece James and the competition I have in that right-back position, but Azpi's professionalism means he's going to be as important for us this year as he always is. He is officially club captain.
"I can't ask players who are as competitive as Azpi to be content with missing one game. Every player here should be keen to play every game.
"I know Azpi is like that. I know him from very well from playing with him.
"I don't expect saints who say, 'okay, I'm not playing this game, everything's fine and rosy'.
"Players need to want to play but, when you have someone like Azpi with the qualities he has as captain, I know he gives everything.
"I know that whether he plays or there might be times when he doesn't he will have the same positive impact on the group off the pitch.
"That's all I can ask for, and it's a pleasure to have him here for that."