Garry O’Connor netted an eighth-minute free-kick for his first goal since September 28 – when Hibs beat St Johnstone for their only home win since February – but Jonny Hayes easily cut through the Hibs defence and levelled after 41 minutes.
The result saw 10th-placed Inverness, who also hit the post through Richie Foran, retain a four-point lead over Hibs, who moved a point clear of bottom club Dunfermline.
How Hibs could do with a positive result in Monday’s Edinburgh derby with Hearts, but they will need to improve on this performance after appearing shaky at the back and lacking a cutting edge up front.
Fenlon’s reign began at Motherwell, when fire forced a half-time abandonment with Hibs leading 1-0, and losses to Rangers, Aberdeen and Dundee United followed.
Hibs took the lead at Tannadice on Saturday and did so again tonight.
Roman Golobart knocked recalled forward Danny Galbraith over as he forged forward, presenting Hibs with a free-kick on the edge of area.
O’Connor, who was returning following a toe injury as one of four changes, stepped up and whipped the ball into the top corner for his 11th goal of the season.
Nine minutes later O’Connor ballooned a 25-yard free-kick over.
The visitors were playing on the break and David Stephens snuffed out a chance for Billy McKay, blocking the striker’s shot as he forged through on goal.
After Hibs failed to clear – their defensive fragility in evidence – Foran curled a thunderous left-foot effort against the post and moments later Hayes shot narrowly wide across goal.
Stephens and Golobart were booked before the visitors equalised four minutes before the interval.
Hayes slalomed through the Hibs defence, beat Galbraith and the space opened up to allow him to hit a left-footed shot into the bottom corner.
O’Connor was booked for simulation after going down in the Inverness box while trying to find an opening. He had been accused of diving to win a penalty in the defeat of St Johnstone and won an appeal against a two-match ban, but the yellow card appeared valid on this occasion.
Galbraith shot over attempting to lob Ryan Esson after a flowing Hibs move involving David Wotherspoon, Leigh Griffiths and O’Connor early in the second half.
Griffiths over-hit an effort soon after as Hibs pushed on, but the hosts appeared vulnerable at the back and Wotherspoon, playing at right-back, was relieved to see Stephens mop up a stray pass across goal.
Midfielder Martin Scott replaced Griffiths in the 63rd minute and was in close attendance when a handball appeal against Kenny Gillet was dismissed following a free-kick into the area.
Greg Tansey came on in place of Lee Cox and Aaron Doran for McKay, with Hibs replacing Richie Towell with Victor Palsson in midfield.
Fenlon put Ian Murray on for Ivan Sproule for the final seven minutes to shore up the defence further as the manager seemed satisfied to settle for a point.
Another appeal for a spot-kick for handball, this time against Golobart, came more in hope than expectation before Sean O’Hanlon and David Davis were booked in quick succession.
Stephens blocked a Gregory Tade effort, Hayes shot well over and Davis had a strike easily saved by Graham Stack as the visitors looked for a winner.
Hibs then had three stoppage-time corners but Inverness held firm as the spoils were shared.