Caley weathered Livingston's early pressure and took the lead in the 16th minute. Barry Wilson was denied by a great save from Roddy McKenzie when the keeper touched his 20-yards hot around the post, but his frustration didn't last long as Ross Tokely rose highest to power a header home from the resulting corner.
The league's bottom side rallied, and Caley keeper Mark Brown had to produce a brilliant save to touch Allan Walker's great shot three minutes later.
Craig Dargo should have made it two in the 31st minute when Tokely teed him up eight yards out, but the striker blazed badly over.
Livi got back on level terms in the 37th minute when captain Richard Brittain was picked out unselfishly by Noel Whelan at the far post and he headed powerfully home.
Dennis Wyness was odds-on to put Caley back in front in the 60th minute when he blocked Harold Pinxten's attempted clearance. He kept his cool and rounded Brown, but was foiled by a great goal-line block by Greg Strong.
Dargo flashed a shot narrowly wide from the edge of the area in the 78th, but it would have been rough on Livi and they went ahead with 10 minutes left when Brittain's pass sliced open the Caley defence for Healy to coolly slot home.
The action wasn't finished as Roddy MacKenzie made another great save to deny Dargo a minute later