"Thanks love u," said the 27-year-old divorcee to her congratulatory fans.
Kroeger's Twitter page made no mention of the wedding, a United Press International spot check indicated. His Facebook fan appreciation page included several posts.
The celebrity human-interest magazine Hello! Canada first reported the engagement with a cover photo of Lavigne sporting a 14-carat diamond ring.
"I knew I was falling for her," Kroeger told the magazine. "It was incredibly powerful and something I'll never forget."
Lavigne, whose 2002 breakthrough single "Complicated," from the album "Let Go," peaked at No. 1 in many countries, said Kroeger, 37, "makes me laugh every day. He takes care of me in every way and is extremely attentive."
Kroeger -- known for songs "Photograph," "Gotta Be Somebody" and "If Everyone Cared" -- proposed with the diamond ring Aug. 8, the magazine said.
But "they haven't set a [wedding] date," a Lavigne friend told People magazine, which followed up on the Hello! Canada report. "They're really enjoying this moment."
This will be Lavigne's second marriage.
In July 2006, she married her boyfriend of two years Deryck Whibley, lead singer and guitarist for the Canadian punk rock band Sum 41.
The marriage lasted a little more than three years, with Lavigne filing for divorce in October 2009.
Whibley and Lavigne continued to work together, with Whibley producing her fourth album, as well as Lavigne's single "Alice," written for the Tim Burton film "Alice in Wonderland."
She started dating Kroeger the following month. The relationship blossomed when they got together to co-write a song for Lavigne's recently completed fifth album.
"They have the same values and really understand each other," a family source told People magazine. "They complement each other and are so happy."