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The Scots were whitewashed last year with five defeats from five matches.
"Clearly they've got to get over the line on at least two occasions," Hastings told BBC Scotland.
"We are playing three away matches but it's not inconceivable that we could win our two home matches against England and France."
The former Scotland captain urged the current crop of players to grasp the opportunity to register a famous win on Saturday in their opening fixture against England at Murrayfield.
"You only get one shot at this opening match against England at Murrayfield," Hastings said.
"I think the last couple of times we've played England up here we've really failed to turn up and I don't expect that to happen on Saturday. I think Scotland can go out there and produce a great performance.
"There was huge disappointment at the end of that World Cup quarter-final match against Australia but they've got to take all the positives into this Six Nations.
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"Vern Cotter has certainly got the players believing in themselves and we've got a talented group of players now, but they've got to go out and do it."
Ahead of his first match since taking charge of England, Eddie Jones suggested that Scotland would be favourites going into the Calcutta Cup clash, despite not having tasted victory over the Auld Enemy since 2008.
Hastings dismissed Jones' comments as mere "mind games".
"I'd expect him to say that, nothing less. Scotland have only won one opening match [against France, in 2006] in the 16 seasons of Six Nations rugby. That's a very stark statistic.
"We've only won three games in the last four years, so to suggest we would be overwhelming favourites would be taking it too far.
"As we're always optimistic at the start of any championship, I'd like to think Scotland can go in believing that they can win the game. That is the most important thing.
"If the players believe that they can go and beat England then they've got a chance. If they don't believe that, it isn't going to happen."
Hastings played a key role in two of Scotland's most famous victories over England - the storied 1990 Grand Slam decider and his debut in the Calcutta Cup fixture in 1986, still Scotland's record victory over England.
"Beating England 33-6 was almost inconceivable, that a Scotland team could beat England by so many points, but we did on that occasion," he recalled.
"If somebody had told me after that game that we'd only beat England once more in my career I'd have thought 'Please no, more than that'.
"John Rutherford was in his absolute element and just played beautifully that day. He was terrific.
"I managed to kick all my goals and that makes a big difference as well. When these things go for you, and it doesn't happen all that often, it's pretty nice when it does.
"We certainly had a good celebration that night."
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Scotland legend Gavin Hastings believes two home victories would constitute a successful 2016 Six Nations for Vern Cotter's side.