In her comments outside the White House on Thursday, Duke had said she was "very satisfied" with the progress made so far.
"I know that it is really a good news story in terms of our ability to reach people and the limited number of deaths that have taken place in such a devastating hurricane," Duke said.
The island has been plunged into a humanitarian crisis since Hurricane Maria hit and the Trump administration has faced a torrent of criticism over slow and disorganized relief efforts.
When informed of Duke's comments during an interview on CNN's "New Day," Cruz appeared shocked and asked host Alisyn Camerota if Duke had really made that statement.
Cruz then called Duke's remarks "irresponsible" and urged the Trump administration to wait until food and water are "in the hands of people" before talking about good news.
"Well, maybe from where she's standing it's a good news story," she said. "When you're drinking from a creek it's not a good news story. When you don't have food for a baby it's not a good news story. When you have to pull people down from their buildings, because - I'm sorry, but that really upsets me and frustrates me."
Cruz continued:
Dammit, this is not a good news story. This is a people-are-dying story. This is a life-or-death story. This is there's a truckload of stuff that cannot be taken to people story. This is a story of devastation that continues to worsen because people are not getting food and water - if I could scream it a lot more louder. It is not a good news story when people are dying, when they don't have dialysis, when their generators aren't working and their generators aren't providing for them. Where is the good news here?
Cruz explained that aid efforts in Puerto Rico have been plagued not by a lack of supplies, but logistical difficulties in distributing food, water, and medicine across the island - particularly in rural areas.
"The good news is that we're getting heard," she said. "The good news is that there are boots on the ground. The good news that the people from FEMA have their heart in the right place and the [Health and Human Services] people know what to do. For heaven's sake, somebody let them do their job."
President Donald Trump on Friday took to Twitter to defend his administration's response to Puerto Rico's crisis. He quoted Governor Ricardo Rossello as saying, "The Administration and the President, every time we've spoken, they've delivered."
Trump went on to say that "big decisions" must be made regarding the cost of rebuilding the island in the wake of the destruction.
Trump is set to visit Puerto Rico on Tuesday.
Watch Cruz's full comments on CNN below:
"Dammit, this is not a good news story" It's "a 'people are dying' story": San Juan mayor slams acting DHS secretary https://t.co/q2mR2utu7k pic.twitter.com/GBx8K1hTW8
- CNN (@CNN) September 29, 2017