The Glasgow giants have reportedly been in talks this week with former Bournemouth boss Eddie Howe.
In the meantime, interim boss John Kennedy is in charge till the end of the season and will have the chance to end a disastrous campaign on a high with a trophy.
The Hoops had to wait until the 56th minute to take the lead through a deflected drive from James Forrest, who was making his first start in six months.
Forrest has missed most of the season with an ankle injury but will be hoping to make Scotland's squad for Euro 2020 with a strong finish to the campaign.
"It's obviously great to start and get the goal," said Forrest. "I'm absolutely delighted after what I've been through for the last few months.