Tansey broke the deadlock against the much-changed visitors when he curled home a 25-yard free-kick just after the hour at the Caledonian Stadium.
Hayes sealed the victory 10 minutes later with a penalty after Matt Doherty handled in the area.
Hibs manager Pat Fenlon rested the majority of his starting line-up ahead of the cup final against Hearts next Saturday.
Eight players who featured in their 4-0 win against Dunfermline on Monday night were switched for counterparts aiming to impress ahead of the Hampden final.
Garry O'Connor, Tom Soares and Matt Doherty retained their places for the final SPL game trip north.
Caley Thistle manager Terry Butcher made a trio of changes to the side who won 1-0 at St Mirren last time out.
Ross Tokely, Thomas Piermayr and Billy McKay replaced Roman Golobart, Josh Meekings and Aaron Doran.
Far from being a meaningless game, Hibs had the opportunity to snatch 10th place and a five figure cash boost in the process.
With around £80,000 extra at stake per league place, it will be a welcome reward for Inverness.
Chances were at a premium in the opening stages with Andrew Shinnie's free-kick hitting the Hibs wall and Ivan Sproule had a shot turned wide by Tokely.
Hayes dragged a shot wide on a rare break from the midfield battle.
George Francomb, moments after being booked, forced the first save of the game.
Ryan Esson was at full stretch to deny the defender's 25-yard low drive.
Inverness responded with Shinnie firing a foot over the bar from just outside the box.
O'Connor was subbed at half-time, perhaps to protect him ahead of next week's final, with Leigh Griffiths coming into the fray.
Inverness threatened with Hayes testing debutant Hibs keeper Paul Grant from 20 yards, but it was saved comfortably.
David Stephens was then booked for a foul on Hayes just outside the box.
Tansey fired home the free-kick from 25 yards to put Inverness ahead on 62 minutes, the the midfielder's fifth goal of the season.
Hayes doubled the lead for Inverness from the penalty spot after 71 minutes. The kick was awarded by referee John Beaton after McKay's cross struck Doherty's arm.
Hayes sent Grant the wrong way as he netted on his 101st appearance for Caley Thistle.
Griffiths' 30-yard angled free-kick was taken easily by Esson as Hibs tried to reduce the arrears.
But the visitors were unable to peg back Inverness, who secured their fifth home win of the season.