The Gunners announced they had secured the 25-year-old on what they described as a "long-term deal" on Thursday, making him the second-most expensive purchase in the club's history, behind Mesut Ozil.
On Monday Newcastle right back Mathieu Debuchy completed a medical ahead of an €11m move to the Emirates Stadium, where he will replace the departed Bacary Sagna.
Gazidis is adamant that Arsenal's acitivity in this summer's transfer market is far from finished.
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"We’ve pulled off a good one with Alexis Sanchez, but there’s more work to be done," the chief executive told Arsenal Fan TV.
When asked if a holding midfielder was his next priority, he replied: "I’m not going to talk about it until it’s done. I’ve got to get back, we’ve got more work to do."
But Gazidis was far more willing to hail the arrival of Alexis, a player he believes is "top class".
It's a really exciting signing for us, we've been wanting to get Alexis into the club and had the opportunity to do it," he told the club's official website. "We're really glad he's committed to us, he had a lot of options and he chose us.
"I think he feels from a football perspective we can give him what he's looking for. It's a great step and takes us one step closer to where we want to get to. We've been really clear, we'll only bring in players that we think really add to the squad and we think he's top class, world class, a great signing for us.
"It's fantastic, it's what it's all about - making sure the club progresses and we can make our fans happy and proud. It's a brilliant feeling when we're delivering trophies and bringing great players in."