Caley Thistle had the better of the game but missed a number of chances to secure their first home victory in eight games with Graeme Shinnie among the most culpable.
Kyle Hutton and Martin Hardie had shots from good positions saved in each half as the visitors failed to maintain their recent goalscoring surge.
With Hibernian stretching their lead over the bottom side with victory at Aberdeen, Dunfermline now need to win at Easter Road on Monday.
Their fate would still be out their hands and with Hibs now eight goals better off, the Pars would probably need Hibs to lose at Inverness on the final day as well as beating Kilmarnock themselves.
Dunfermline had revived their survival fight with seven goals in two games and they welcomed back full-backs Austin McCann and Jordan McMillan back from illness and injury. Nick Ross and Kenny Gillet replaced Owain Tudur Jones and the suspended Jonny Hayes for Inverness.
The game was preceded by a minute's silence for Inverness fan John MacDonald, who collapsed and died during his team's defeat at Rugby Park on Saturday.
The home side had the better of the first half hour. Billy McKay came close to getting a touch on Aaron Doran's ball across the goalmouth following a good run from the Irishman.
Chris Smith produced a good diving save to stop Steve Williams heading home Doran's corner and the Pars goalkeeper then held McKay's 20-yard drive.
Inverness claimed for a penalty when Doran went down under McCann's challenge after the left-back had misjudged a long ball.
Inverness manager Terry Butcher, still angry over Hayes' red card and a denied penalty claim at Kilmarnock, had advocated the use of professional referees in his programme notes and he got no joy on this occasion from Stevie O'Reilly, who gave a goal kick.
Dunfermline had been restricted to a weak David Graham effort from 25 yards but they had a decent chance in the 34th minute when Joe Cardle's cross found Hutton. However, the on-loan Rangers player volleyed straight at Ryan Esson from 12 yards.
Inverness soon had another scare when McMillan's powerful cross skimmed off the head of Gillet and not far over his bar.
With Hibernian two goals up at half-time, there was need for more urgency from both teams and the game stepped up a gear.
The Inverness fans displayed their frustration when Doran passed up a chance to shoot from 12 yards before Shinnie failed to catch hold of a volley.
Inverness survived a couple of goalmouth scrambles and almost struck on the counter-attack.
Greg Tansey's 50-yard pass got McKay round the side and he laid the ball back for Shinnie but the midfielder scuffed the ball just wide from 18 yards.
Richie Foran chipped over after good skill by McKay and Dunfermline manager Jim Jefferies brought on Hardie for Gary Mason in a bid to inject more attacking threat.
Doran slid Tansey's driven ball just wide at full stretch after McKay had got in behind again. Andrew Shinnie replaced his brother for his first appearance since breaking a foot in January. The home side almost paid for their missed chances when Alex Keddie met Cardle's inswinging free-kick but the defender's header landed on the roof of the net.
The unmarked Hardie had a great chance with 11 minutes left when a defensive header fell for him 18 yards out. The midfielder connected well with a half-volley but the shot was straight at Esson and the goalkeeper clung on to the ball at the second attempt.
Jefferies brought on forwards Andrew Barrowman and Liam Buchanan but Dunfermline could not get the ball forward in the closing stages.