Wednesday are four points adrift of safety with two games remaining, and need at least a point against Nottingham Forest next weekend to have any chance of going to relegation rivals Derby on the final weekend with their survival hopes still alive.
Josh Windass, who scored a first-half penalty, and Jordan Rhodes both missed excellent second-half chances at the Riverside, with goals from Yannick Bolasie, Josh Coburn and Duncan Watmore securing Middlesbrough’s success.
Smith, who is standing in while Wednesday manager Darren Moore recovers from pneumonia, accepts his side are in deep trouble, but is adamant no-one will be giving up while there is still a chance of clambering out of the bottom three.
Smith said: “We’ve still got a chance. Mathematically, it’s still possible, so we will keep fighting.
“It’s must-win in the next game, but, to be honest, it’s been like that for the last few weeks. We need to get three points in the next game – it’s as simple as that. Whatever it takes, we have to go and get three points.”
Smith felt his side let themselves down with some defensive lapses at the Riverside, with two of Middlesbrough’s goals coming from set-pieces.
Smith said: “We’re obviously disappointed with the result – we felt we should have got something from the game. We’re disappointed with the goals we’ve conceded – a lack of concentration has cost us.
“Players have to take responsibility to not switch off from set-pieces. They have to be focused and be ready. We’ve generally been well-organised from set-pieces, but today wasn’t like that.”
Middlesbrough manager Neil Warnock was delighted to see 18-year-old Coburn claim his first senior goal as he headed home from Neeskens Kebano’s cross.
The teenager only made his senior debut when he came off the bench at Rotherham in midweek and he had only been on the field for eight minutes against Wednesday when he rose to score. Warnock said: “He’s done really well in training and worked his socks off. He took his goal well. When you look at it again, it’s not an easy one. It’s a really good header.
“He deserves it. The lads gave him a round of applause and signed his shirt. He’ll be a happy lad tonight like Connor (Malley) was the other night.
“I don’t think we can rush these young lads. I was going to put Josh on a bit earlier but Blackie (Kevin Blackwell) said it’s a bit physical. But 10 minutes later I thought, ‘Sod it, there’s nothing to lose’.
“He battled well though. He won all of his headers. It wasn’t just his goal, was it? He was determined and you could see how much desire he has.”
Boro completed a quickfire South Yorkshire double after triumphing at Rotherham on Wednesday night, by making the most of some poor Owls defending.
Bolasie was a constant threat as he drifted across the frontline, and while he failed to find the target with a low effort in the sixth minute, the Everton loanee claimed the opener 14 minutes later.
Julian Borner’s poor first touch enabled Bolasie to win possession, and after advancing into the right-hand side of the area, the forward drilled a low shot into the bottom corner via a slight deflection off Tom Lees.
George Saville headed over as the home side continued to press, but Wednesday hit back and Andre Green’s deflected effort whistled just past the post before Borner spurned an excellent opportunity when he hooked wide from eight yards.
The Owls struck six minutes before the break. Green was tripped by Marc Bola and with Barry Bannan having been removed from penalty duties after his miss against Bristol City, Windass stepped up to fire a low finish past Jordan Archer.
Windass hit wide four minutes after the interval and while Wednesday dominated possession for most of the half, they fell behind in the 73rd minute to Coburn’s header before Boro made the points safe from another corner with nine minutes left through Watmore.
Middlesbrough: Archer, McNair, Hall, Bola, Kebano (Spence 79); Howson, Saville, Johnson, Bolasie (Malley 89), Akpom (Coburn 65); Watmore. Unused substitutes: Robinson, Bettinelli, Hackney, Kokolo.
Sheffield Wednesday: Westwood, Hutchinson, Lees (Urhoghide 45); Borner (Hunt 82), Palmer, Pelupessy, Bannan, Reach, Windass; Paterson (Rhodes 71), Green (Harris 82). Unused substitutes: Penney, Dele-Bashiru, Shaw, Wildsmith, Kachunga.
Referee: J Gillett (Australia).
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