'It's a big blow for me,' said Zola, whose side miss out on a quarter-final tie at Everton a week on Sunday. 'Everyone knows how important winning the Cup can be and how special this competition is to me. 'I wanted my players to have the same pleasure that I had a few years ago. It was our ambition to win this competition.
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'I'm sorry and disappointed for our supporters because they were fantastic and they came here in big numbers and we wanted to give them something more. It has not been a great performance but the players gave everything which was very encouraging.'
Stewart Downing put Boro ahead in the fourth minute with a brilliant 25-yard free-kick, curling his effort beyond the wall and leaving England colleague Robert Green flailing as the ball soared into his top left-hand corner after debutant Radoslav Kovac fouled Tuncay Sanli.
An invigorated Tuncay doubled the lead in the 19th minute when he volleyed past Green from 20 yards after James Tomkins' attempt to intercept Gary O'Neil's cross fell invitingly to the Turkish international.
Zola's defence was at sixes and sevens and the 4,000 travelling supporters tempted to Teesside by their club's offer of free transport made their dissatisfaction known by booing their side off at the interval.
'It wasn't our best performance but it is not easy when for the second time in a few days we start with an opposing player putting the ball into the top corner early on and scoring another straight away,' Zola said.
'That would make it difficult for anyone, especially in terms of confidence. We needed to score early in the second half but we didn't. Their confidence went up and ours went down a little.
Not even Boro's biggest killjoys would have been complaining about the noise levels because their supporters finally had something to shout about.
The curmudgeons had left their fellow 'Smoggies' gobsmacked when they were warned to keep the noise down by club officials because of complaints that 'persistent standing and the constant banging and noise' was 'driving some fans mad'.
Watched by Everton manager David Moyes, the home fans raised the roof and West Ham raised their game after half-time but were kept at arm's length by Boro, apart from brief scare when substitute David Di Michele sent Scott Parker clear but his shot missed the target by inches.
Boro manager Gareth Southgate, whose side face Liverpool on Saturday, said: 'The quality of the early goal probably knocked the stuffing out of them. Their front two of Tuncay and Jeremie Aliadiere set the tone because they worked their socks off for the team.
'This keeps people's dreams alive. That's why we always wanted to take the Cup seriously. There are some good players here – it's just a state of mind. If we get our good players playing well we have the quality to stay up.'
Downing said: 'The important thing is the result and breeding confidence for Saturday. That kind of performance has been coming.
'Keep playing like that and we'll have no worries and we'll not fear anyone. We have to do it in the league now.'
FA Cup quarter-final draw
Coventry v Chelsea Fulham v Manchester United Arsenal or Burnley v Sheffield United or Hull Everton v Middlesbrough
Ties to be played on the weekend of March 7-8.