The 31-year-old had to wait until the loan window opened before making a move, with incoming Leeds owner Massimo Cellino having blocked a deal in January.
Varney admits the turmoil at Elland Road left him sweating on a transfer, but he is looking forward to relighting his footballing spark at Ewood Park.
He said in the Lancashire Telegraph: "I'm delighted to finally be here.
"I've had a lot of conversations with the club and the manager and I'm happy to get it over the line after a week of turmoil at Leeds.
"Once it was made aware that I might not get a new contract at Leeds they stopped playing me, so I understood I would have to move on and I was delighted when I heard of the interest from Blackburn.
"I would have come as soon as I could but obviously the turmoil at Leeds stopped me. I'm just really pleased to be here now.
"I can put all of what's happened behind me and use that, not anger, but there's a lot inside that I need to get out and I'm hoping to do that on the pitch for Blackburn.
"The gaffer has given me a great chance and I'm hoping to repay that faith soon."
Varney, who will be out of contract in the summer, added: "It's been well documented that I've been trying to get out of Leeds for a while.
"I lost my motivation at Leeds and I'm a very positive person, so for me to do that shows something had gone wrong."