Boro will now visit Everton on Saturday March 7th.
Gareth Southgate's team have found scoring difficult in the Premier League with only one goal in their last nine games.
Yet they hit West Ham with a two-goal salvo in the opening 19 minutes from which the visitors never recovered.
It was a pity an impressive Middlesbrough showing was watched by 15,602, the lowest crowd of the season at the Riverside Stadium. West Ham were well supported by several hundred fans.
But the followers who had made a 500-mile round trip to the north-east must have been bitterly disappointed by the performance of their team.
Middlesbrough created far more scoring chances while West Ham posed only a limited threat in attack.
The opening goal by England winger Stewart Downing could qualify for a FA Cup goal of the month contender after Radoslav Kovac had brought down Sanli Tuncay.
Downing fired in a stunning 25-yard free-kick high into the corner of the net. The power and accuracy gave Robert Green no chance in goal. This was only Downing's second goal of the season and, iIronically, his first was in the cup tie at Upton Park.
Downing followed up with a shot just wide of the far post following a defensive lapse by Matthew Upson.
Middlesbrough went further ahead after 19 minutes with a goal by Turkey international Tuncay.
Tuncay has been out of form in recent weeks but he gave Middlesbrough fans something to cheer with a shot from outside the penalty area.
James Tomkins attempted to clear a right-wing cross by Gary O'Neil but he only put the ball in the path of Tuncay.
Kovac fired in a strong shot from outside the penalty area for the visitors but it was easily saved by Bradley Jones. This was West Ham's first goal attempt after 21 minutes.
When Herita Ilunga missed the ball in the penalty area, O'Neil prepared to pounce but Green got to the ball first.
West Ham's Carlton Cole fell awkwardly in a Middlesbrough attack. He looked in pain with a knee injury but recovered after treatment.
Tuncay blasted a shot over the bar after Downing had pulled the ball back to him.
West Ham manager Gianfranco Zola must have been bitterly disappointed with the performance of his team in the first half.
In the second half of a game easily controlled by referee Steve Bennett, there was never any real prospect of West Ham pulling the game round.
Jubilant Middlesbrough fans chanted "We're going to Wembley." That may be over-optimistic but at least the win should boost confidence for their Premier League survival fight.