Kejriwal, the Delhi chief minister who has set his eyes on the western state, said it was the "triumph" of AAP's fight against corruption in Gujarat.
AAP spokesperson Ashutosh said Patel's offer to resign was a mere drama and that she was actually being "sacked".
"Anandiben's resignation is the outcome of AAP's soaring popularity in Gujarat. Bharatiya Janata Party is badly scared in Gujarat. Her resignation is also the triumph of AAP's fight against corruption in Gujarat," Kejriwal said in a series of tweets.
The AAP national convenor has visited Gujarat twice in the recent past. He had met the dalit men who were flogged for skinning a dead cow.
In another tweet, he took a swipe at the prime minister, saying people were dying to hear him on a host of contentious issues. "Sir, pl speak on dalit atrocities, gau rakshks, kashmir, akhlak, farmer suicides, daal price. Ppl dying to hear u on these (sic)."
Ashutosh said BJP was forced to change Patel as chief minister and that it was looking for a new leader as a "scared" Modi knows he is "losing in his home turf".
"Anandi Ben is not resigning. It is drama. She is being sacked," Ashutosh said. He also put out a series of tweets saying Modi is rewarding Patel "with governorship".
"BJP has to answer why Hardik (Patel) was in jail for treason? Why thousands of Patel women/kids were beaten in homes? BJP has to answer why dalits were beaten in Una, Rajula so brutally? Why Gau Rakshak Dal is roaming free? Why false cases against dalits?" he said in a series of tweets.
Patel has offered to resign requesting the BJP leadership to relieve her from the post as she is soon going to be 75.
The BJP for the first time is facing a tough challenge from opponents in Gujarat since Narendra Modi moved to the Centre.