West Ham's 4-4-2 formation, with Freddie Sears starting his first match in over three months and Radoslav Kovac making his debut in central midfield, simply encouraged the hosts to flood forward.
And when Kovac felled Tuncay on the Middlesbrough right, 25 yards from goal, Downing stepped up to send a magnificent effort past Robert Green via the angle of post and crossbar.
Boro went in search of a second and Downing zipped a shot narrowly wide after Matthew Upson's suicidal cross-field pass was intercepted.
O'Neill then sent a fizzing aerial ball forward and James Tomkins succeeded only in hooking his clearance to the on-rushing Tuncay, who fired convincingly past Green.
Backed by a healthy travelling support, which was bussed up to Teesside by the club, West Ham eventually tested Boro goalkeeper Brad Jones when Kovac shot low following Herita Ilunga's burst into the penalty area.
Attempting to stay true to their manager's philosophy, West Ham showed their ability to put together fluid passages of play and Hoyte was forced to make a vital interception as Kovac primed himself to convert Carlton Cole's centre.
Boro came close to extending their advantage before half-time as O'Neil's deep cross from the right was played back by Downing, only for Tuncay to blast high over the bar.
With West Ham desperate to work their way back into the match and Boro hungry for more goals, the second period got off to a pulsating start.
Aliadiere saw an effort deflected narrowly wide and Green caught well to prevent O'Neil scoring after a slick exchange between Aliadiere and Tuncay.
At the other, Sears and Mark Noble came close to breaking through Boro's impressive defensive wall and, having introduced Jack Collison and Di Michele, West Ham continued to probe.
The visitors might have been level on the hour mark had Scott Parker's left-footed drive from Di Michele's throughball not squirmed just wide of the post.
Parker's influence began to grow but, with Boro all too willing to get forward when in possession, West Ham could ill afford to neglect defensive duties.
Green foiled Aliadiere expertly when the Frenchman was played through by O'Neil, Matthew Bates saw a long-range drive tipped over and Tuncay's deflected shot ran marginally wide.
But those misses proved insignificant and Boro now travel to Goodison Park with a place in the semi-finals and a trip to Wembley at stake.
Middlesbrough manager Gareth Southgate: "There was a good feel about the ground tonight and everybody has gone away with a good feeling.
"It's important because football is about dreams and we're keeping the dreams of our supporters alive.
"We've got a tough a touch draw in the next round but FA Cup quarter-finals away from home are not a bad thing from my experience.
"It's something to look forward to, it keeps some enjoyment in the season and gives us some confidence for the league."
West Ham manager Gianfranco Zola: "Middlesbrough were very good and, honestly, they deserved to win.
"It's a pity because so far in the FA Cup we have done well and we had many supporters here.
"We wanted to give them more satisfaction, which they deserved. It hasn't been the case and we're very happy about that.
"We've slowed down a little bit but it's understandable and normal at this stage of the season. We are very determined to start again and setting up another run."
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