Alves' 18-yard strike was therefore against type when it came on the stroke of half-time - and play barely picked up after the break.
Referee Phil Dowd did not help matters, the game barely allowed to get going at all thanks to his officiousness, although he was right to delay the match following a horrid clash of heads between Kieron Richardson and Didier Digard, the former needing stitches and the Boro midfielder eventually taken off with concussion.
Still, as the game wore on Sunderland began to press more and more, and in a frantic finale it was they and not the hosts that looked capable of pulling off victory.
Sbragia pleased with point
First Jones slid in his crucial equaliser from six yards after Healy - who had only been on the pitch two minutes - had picked him out brilliantly from the right.
Then Johnson came to Boro's rescue with a stunning backheeled clearance off the line after Edwards had headed Andy Reid's corner goalwards.
And finally only a brilliant last-ditch tackle from Tony McMahon denied Cisse after the Sunderland striker had been put through clear on goal on the stroke of full-time.
All that meant that Boro were the happier of the two sides when the full-time whistle blew, even though the draw means they are without a league win since 9 November and now sit just a point off the relegation zone.
Middlesbrough manager Gareth Southgate:"We'd have liked to have got more out of the game but a draw's about fair. We had the better of the first half and scored a great goal but a couple of injuries meant we lost momentum a bit and we were hanging on in the end.
"We needed a great clearance from Adam and a fantastic tackle from Tony to save us and, as I always say, if you can't win make sure you don't lose, so from that point of view I'm happy.
"As for Downing - his performance was a real highlight. He's accepted our stance and totally understands it. He's up for the challenge of staying here beyond this month and helping us remain in this division - that's the end of it, the whole saga is done and dusted."
Sunderland boss Ricky Sbragia:"I'm definitely happy with a point, especially after being 1-0 down.
"We got caught out on the stroke of half-time so at the break we asked them to up the tempo and the longer the game went on the better we looked. We could even have won it, but we're happy.
"I don't know if Edwards' header was over the line or not - I don't think it's even clear on the replays - but the officials didn't give it anyway, and we accept that."
Middlesbrough: Turnbull, McMahon, Riggott, Wheater, Pogatetz, Sanli, Digard (Bates 65), Walker (Johnson 47), Arca, Downing, Alves (Mido 86).Subs Not Used: Jones, Taylor, Emnes, Huth.
Goals: Alves 45.
Sunderland: Fulop, Chimbonda, Nosworthy, Ferdinand, Collins, Malbranque (Edwards 54), Whitehead, Richardson (Healy 80), Diouf (Reid 67), Cisse, Jones.Subs Not Used: Colgan, Tainio, Murphy, Yorke.
Booked: Malbranque, Richardson, Nosworthy, Jones.
Goals: Jones 82.
Ref: Phil Dowd (Staffordshire).
BBC Sport Player Rater man of the match: Middlesbrough's Afonso Alves 7.76 (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.Player Rater