Hibs enjoyed the majority of possession on Saturday and Victor Palsson had a header blocked on the line in the first half before Booth lofted a volley over Ryan Esson and into the visitors' net after 58 minutes and substitute Stevenson struck a deflected second after 89 minutes.
Adam Rooney twice came close for Inverness, but each time was denied by Hibs goalkeeper Graham Stack.
Booth made his Hibs first-team debut in last month's Scottish Cup fourth-round replay loss at Irn-Bru Second Division Ayr.
How Hibs fortunes have changed since.
The end of the transfer window, which saw manager Colin Calderwood recruit six players, has coincided with an upturn in fortunes and today's result saw Hibs claim a 100% record from four games in February.
With six games remaining until the SPL split a top-six place may yet be the target, but they go into Tuesday's match with bottom club Hamilton still in 10th place.
Inverness remained in sixth place despite the loss and now lead their Easter Road rivals by five points in a congested table.
Four of Calderwood's six January signings began the game in an unchanged starting XI, while Chris Hogg, the former club captain at Easter Road, was named in a Caley Thistle side showing two changes from the 2-0 win over St Johnstone, with Eric Odhiambo and Grant Munro restored to the starting line-up.
While Hibs' confidence has soared of late, Inverness have enjoyed a strong campaign so far and had previously been beaten just twice away from home.
The visitors had the first opportunity when striker Rooney, a rumoured target for Calderwood last month, attempted to lob Stack from an acute angle but saw the effort held.
Akpo Sodje, a forward Calderwood did manage to sign last month, shot narrowly wide at the other end before Martin Scott, another new recruit, failed to test Esson.
Hibs then twice came close to opening the scoring.
Esson saved a David Wotherspoon shot after a flowing Hibs move and Derek Riordan's follow up effort went narrowly wide.
Having almost gone ahead, Hibs nearly found themselves behind as Inverness raced up the other end.
Richie Foran hit a fine cross from the left into the six-yard box and Rooney was denied as Stack saved with his knees.
Riordan then went close with an inswinging shot from just inside the box which Esson had to turn around the post for a corner.
Booth's resulting set-piece found Victor Palsson six yards out, but his header was blocked on the line by Russell Duncan, who was guarding the far post.
Riordan fired a long-range shot straight at Esson early in the second period as Hibs resumed on the attack.
Stack was then called into action again, narrowing the angle and spreading himself to block a Rooney shot from 10 yards.
After four half-chances, Hibs went in front after 58 minutes.
Hogg conceded a soft throw-in and it was taken quickly by Riordan to Booth.
The 19-year-old left-back lifted the ball on the volley over Esson and into the back of the net for his first Hibs goal.
Esson saved from Riordan before Ross Tokely fired well off target at the other end when the ball dropped to him following a corner.
Hibs struggled to find an opening for a second before a half-cleared ball dropped for Stevenson and his deflected effort found the net from 25 yards to seal victory.