Villa have won two of their last three top-flight games to move into the relative safety of 16th and three points above the drop zone, while also reaching the FA Cup semi-finals.
And Bosnich, who made over 200 appearances for Villa and won two League Cup titles before joining Manchester United in 1999, claimed that not only should Villa avoid the drop - they should be challenging for top-six honours.
"Of course they can [avoid relegation]. The bottom line is Paul Lambert did the right thing [by resigning before Sherwood's arrival last month] because they were going nowhere," the ex-Australia keeper told Perform.
"They are a club that should be in the top six. They are a club that has won the European Cup and they are the biggest club in the Midlands.
"They deserve to be up in the top six, not struggling for relegation. The Villa fans are superbly loyal, very good fans.
"Sherwood is the epitome of what Aston Villa is about. He has great passion and far greater knowledge than what a lot of people give him credit for.
"And we saw in the FA Cup game the type of passion required. Unfortunately they had a loss to Swansea but I believe it was the right move at the right time."
Villa will face either Blackburn Rovers or Liverpool in next month's FA Cup semi-final, but Bosnich has warned Sherwood to heed lessons from Wigan Athletic - who won the Cup but were also relegated to the Championship in 2013.
"It is a wonderful thing but it should not take away from the focus that the most important thing is staying in the Premier League," he added.
"You don't want to end up like Wigan."
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.