As so many teams have found this season, Fulham are resolute and well-drilled defensively and after weathering the initial storm the Cottagers sprang to life after 25 minutes when Bobby Zamora was denied by a last-ditch dual challenge by David Wheater and goalkeeper Brad Jones.
Johnson came even closer moments later when he cut in and unleashed a curling effort that deflected off Wheater's boot, beat Jones but hit the inside of the right-hand post and dribbled along the goal-line before being cleared.
Following their lucky escape, Middlesbrough were re-energised by Gareth Southgate's half-time team-talk and forced Schwarzer into three good saves in a 10-minute spell.
Hodgson pleased with Boro draw
Tuncay and Downing were both denied by the Australian, who then tipped over a shot from Tony McMahon from a tight angle on the right.
Again though Fulham withstood Middlesbrough's initial assault and slowed the pace of the match down to frustrate the home side.
Middlesbrough's fans rose in unison with 20 minutes left to claim for a penalty after a Downing shot had ricocheted onto Dickson Etuhu but referee Peter Walton decreed that it was ball to arm.
Schwarzer targets Europe after draw
As the game entered the final 10 minutes, Boro understandably committed men forward leaving space at the back for Fulham to exploit and the visitors almost capitalised when American Clint Dempsey found Zamora running in at a slight angle from the right but he blazed his shot woefully over the bar.
Dempsey could then have won it for the visitors but his flick was saved by Jones before Zamora's follow-up shot was blocked by Matthew Bates.
Middlesbrough manager Gareth Southgate: "I'm proud of the way the players have played today as we knew it would be a tough game.
"It wasn't a performance I can be too critical of. Every player played with everything they had, with passion and with heart.
"We created a few chances but unfortunately we were unable to convert them.
"We're still in their fighting and we know it's a mountain to climb but we're taking little steps. We're going to have to win a game people don't expect us to."
Fulham manager Roy Hodgson: "I have said many times that the UEFA (Cup) has never been an actual target for us.
"But of course, if we continue to play well, maybe things will fall into place and maybe we will get a European spot.
"If we do, we will be very grateful and we will do our best to represent England in the right way."
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