Everton manager Moyes was somewhat vexed on the touchline by the time fourth official Ben Toner indicated six minutes of added time at the end of the first half, but his mood would soon improve.
His team had created by far the better opportunities against Marco Silva’s side, but a combination of wayward finishing and the linesman’s flag had kept the Cottagers on level terms.
Summer signing Barry, who retained his place in Moyes’ starting line-up despite a glaring miss against Sunderland on Monday, is desperate for a first Premier League goal so would have been cursing his luck for having an effort chalked off after Ndiaye strayed offside from Gueye’s low cross.
French striker Barry should have seized an earlier chance when he rose unmarked to meet Dewsbury-Hall’s cross earlier in the half, sending his header high over the bar.
Defender Tarkowski, who had a close-range volley disallowed, was denied by the crossbar not long after that, but Fulham were unable to clear their lines and Gueye pounced to finally put the home side in front.
The second half would have been far more comfortable for Everton had Dewsbury-Hall’s sweeping finish not been ruled out, but the Toffees needed Pickford and Ndiaye to preserve their narrow advantage with key interventions as Fulham began to commit more men forward.
Keane’s late effort, which appeared to come off his shoulder, settled the home fans’ nerves and earned Everton their first win since beating Crystal Palace over a month ago.