Boro winger Adam Johnson found space on City's right flank and found Alves, who from six yards out saw his low shot superbly kept out by Given.
Minutes later, Shaun Wright-Phillips's 18-yard shot was deflected into the path of Bellamy, who like Alves earlier, saw his effort well kept out by Jones.
Alves then tested the Given with an 18-yard swerving shot that turned at the last minute, but the keeper reacted well to keep it out.
Ireland, who was easily City's most consistent player, then headed Wayne Bridge's cross against the bar before Given produced yet more heroics saving at the feet of Alves.
Bellamy was a constant threat in the Man City attack
It took until the 50th minute for the deadlock to be broken and it came from the lively Bellamy.
The striker found space on the edge of the area before unleashing a fierce effort that Jones had no chance of keeping out.
Alves was then denied for a fourth time by Given, with the Irishman diving full stretch to keep out the Brazilian's 20-yard driven shot.
It was Middlesbrough's last meaningful effort of the game with City holding on for the victory despite a late penalty appeal.
Southgate must now hope he is given time to turns things around at the relegation threatened club who have the relief of an FA Cup visit to West Ham next weekend.
Manchester City manager Mark Hughes: "Shay Given at times had to produce what he is capable of. I'm delighted with the way he has been able to come in and hit the ground running.
Hughes applauds Given debut
"If you look at our back four, it's quite light in terms of Premier League experience and that's one of the main reasons I brought Shay in. We benefited from his experience.
"I think on the balance of play we deserved to win. We created good chances and took the one that won us the game."
Middlesbrough manager Gareth Southgate: "We made probably one error as a team defensively and we've been punished for it.
We need to start taking chances - Southgate
"We need to take our chances - it's as simple as that. We created chances and would have hoped to have scored at least one of them.
"Even after going a goal behind there was no sign of anyone throwing the towel in. No-one is hiding. We played well and their goalkeeper played very well, and that's what got them the points."
Man City: Given, Richards, Onuoha, Kompany, Bridge, Zabaleta, De Jong, Ireland, Wright-Phillips, Bellamy, Robinho (Caicedo 83). Subs Not Used: Hart, Elano, Vassell, Garrido, Fernandes, Evans.
Goals: Bellamy 51.
Middlesbrough: Jones, Wheater, Riggott (Hoyte 68), Huth, Pogatetz, Adam Johnson (Sanli 74), Bates, Digard (O'Neil 60), Downing, Alves, King. Subs Not Used: Turnbull, Emnes, Shawky, Arca.
Booked: O'Neil.
Att: 40,588
Ref: A Mariner (Coventry).
BBC Sport Player Rater man of the match: Manchester City's Shay Given 8.53 (on 90 minutes).
Please note that you can still give the players marks out of 10 on BBC Sport's Player Rater after the match has finished.
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