The Danish midfielder is playing for his future at Fulham by trying to earn a new contract at the club.
His current deal expires in the summer and he went someway to justifying his case with the only goal in a dogged display from the Londoners.
Meanwhile, David Moyes' side, and in particular Andy Johnson, must be left questioning how they did not get at least a point form a match they dominated.
Everton started the brightest with Johnson looking lively in attack. His pace almost lead to the opener in only the second minute, but he was out-muscled by Zat Knight who calmly swatted him away.
Johnson was again causing problems for the Fulham defence when his teasing cross almost dipped under the crossbar and forced Antti Niemi to make a save.
At the other end, Joleon Lescott needed to be on guard to clear Tomasz Radzinski's cross, which looked to be heading straight for Brian McBride at the far post.
Johnson should have given Moyes' side the lead in the 13th minute when Fulham made a hash of Phil Neville's long throw.
The ball found its way through to Johnson, who miscued his acrobatic kick but Niemi needed to make the save.
Three minutes later the former Crystal Palace striker saw his appeal for a penalty turned down by referee Martin Atkinson as Everton took control.
Johnson was clearly caught by defender Ian Pearce and proceeded to fall in the box, but Mr Atkinson ruled there was nothing sinister in the challenge.
Leon Osman then fell foul of a decision from the officials when he was judged to be offside. The young winger broke the Fulham offside trap and shot at goal but Niemi pulled off an excellent save nonetheless.
Everton pressed forward and a mistake by Luis Boa Morte, who was making his first start since September after recovering from a fractured cheekbone, nearly let in Mikel Arteta.
The 24-year-old Spaniard raced on to the loose ball after Boa Morte had slipped but was denied a clear shot at goal by an excellent recovery tackle from Liam Rosenior.
Johnson then wasted the best chance of the half three minutes before half-time when he somehow managed to fire over the bar from only two yards after Tim Cahill had supplied the cross.
Ironically, after Everton had dominated the whole of the first half, Fulham had their first shot at goal when McBride turned in the box and shot low, but Tim Howard made an easy save.
After the restart Fulham looked a different side as they took the game to Everton.
Pearce led the way when he almost headed in from Jensen's free-kick but was denied by Howard.
Jensen then struck the killer blow in the 66th minute. The Danish midfielder picked up the ball from a throw in before turning inside and surging towards Howard.
Jensen was strong enough to hold off the challenge from Cahill and unleashed a curling effort which deflected off Lee Carsley and into the net.
Jensen, who was struck by a coin thrown from the Everton fans, almost doubled the lead moment later when he tried his luck with a free-kick from 25 yards which fizzed past Howard's post.
Moyes introduced James Beattie to improve his striking options, but it was Johnson who was again looking likeliest to put them back on level terms.
He raced through the Fulham defence but was stopped by two excellent saves from Niemi.
Despite a late surge of pressure from Everton, Fulham clung on to take all three points.