Bevvy will leave the Warriors at the end of the season after agreeing a two-year deal with Bristol.
And the scrum-half believes fate will help the Scots see off Saracens tomorrow at Watford and secure a last-four spot in the Challenge Cup.
He said in the Daily Record: "Sport often throws up amazing coincidences and I have a gut feeling about what is going to happen over the next few weeks.
"I'm convinced we'll earn a positive result at Vicarage Road this weekend and Bristol will come up to Hughenden to battle it out for a place in the final."
Bevvy says he still has plenty of work to do at Glasgow before moving south.
"There are so many exciting things to focus on at Glasgow. We've had a good season but we can turn it into a great season.
"Not only are we in with a shout of becoming the first Scottish team to win a European trophy, we have six very important Magners League matches left with the prospect of achieving our best-ever finish.
"Only when all that is out of the way will I start to concentrate on my move down south."
Bev added: "We will travel with cautious confidence. We've played against them twice in the sectional phase so we know what to expect.
"Saracens are a very strong side and beat us at Vicarage Road the last time but we are definitely more powerful in the scrummaging department now.
"We proved we can score tries against them in that 28-23 defeat and the only reason we lost the game was because of two pushover scores."