The Second Division leaders stunned Hibs by taking the lead after just 15 seconds of the fourth-round clash through Greig Stewart, but goals from Leigh Griffiths and Eoin Doyle gave the visitors a half-time lead and another from David Wotherspoon put them 3-1 up shortly after the break.
A superb Jon Robertson goal midway through the second period gave Colin Cameron's side hopes of a comeback but despite waves of pressure late on they could not find an equaliser.
After four defeats and a draw for Fenlon since he took the reins in November, the victory will have come as welcome relief after their Premier League troubles.
The home side got a dream start after scoring with their first attack.
The Blue Brazil took the kick-off and Cameron fired a pass up to the edge of the area where Greig Stewart benefited from the break of the ball off Easter Road defender Sean O'Hanlon before curling a superb shot beyond the despairing dive of Mark Brown.
The SPL men were clearly shocked by the early breakthrough but responded in impressive fashion.
Hibs dominated possession and Griffiths wasted a great opportunity to produce the perfect riposte when he shot wide from 14 yards out after just two minutes.
However, the Wolves youngster, who extended his loan deal to the end of the season on Friday, grabbed a spectacular equaliser in the 18th minute.
Cameron was robbed of possession by Lewis Stevenson and Griffiths fired a magnificent shot in off the underside of the bar from 30 yards out.
Two minutes later, home keeper Thomas Flynn - a former Hibs youth player - was called into action as defender John Armstrong slipped and presented Doyle with a chance on the left edge of the area.
However, the young shot-stopper was equal to the Irishman's curling shot with a dive full-length to his left.
Flynn again saved from Doyle in the 26th minute as he raced onto Callum Booth's ball over the top, but the striker was not to be denied just seconds later as Hibs deservedly took the lead.
Michael Hart burst forward down the right and found Doyle in the inside-right channel. Flynn kept out the striker's first effort but Doyle made no mistake from the rebound.
The Easter Road side could not have feared a worse start to the match but they could not have hoped for a better response.
Griffiths then had a curling free-kick tipped over by Flynn and full-back Kenny Adamson popped up on the line with a crucial intervention as Doyle looked to have nodded O'Hanlon's header in from close range.
Booth did just enough to put Stewart off soon after the interval as O'Hanlon allowed Adamson's through-ball to skip through, before Hibs gave themselves a two-goal cushion in the 54th minute.
Danny Galbraith's deflected cross after a short corner worked with Ivan Sproule hung up for Wotherspoon and the midfielder planted his diving header past Flynn from six yards out.
Cowdenbeath looked deflated but the tie took on a completely different complexion midway through the half as the home side pulled a goal back.
Robertson showed great poise as he latched onto a chipped pass, lifted the ball over marker O'Hanlon and volleyed superbly past Brown.
A sense of panic appeared to spread through the worried visiting ranks but, with Robertson rifling over and curling a free-kick just wide, there was not to be a late equaliser for the home side.
Cowdenbeath Flynn, Cowan, Armstrong, Mbu, Adamson, McKenzie, Ramsay, Robertson, Cameron, Linton (Coult 56), Stewart.
Subs Not Used: Bejaoui, Brett, Winter, Milne.
Goals: Stewart 1, Robertson 69.
Hibernian Brown, Hart, O'Hanlon, Hanlon, Booth, Sproule, Wotherspoon, Stevenson, Galbraith (Towell 80), Griffiths, Doyle (Sodje 88).
Subs Not Used: Grant, Stephens, Scott.
Goals: Griffiths 18, Doyle 27, Wotherspoon 54.
Att: 2, 670
Ref: Bobby Madden (Scotland).