Middlesbrough secured a victory that moved them two places up the table as Portsmouth failed to score for the sixth consecutive league match at home.
Tuncay's strike was enough to hand Boro all three points
Pompey started briskly but failed to create a clear opening and paid for it on 20 minutes when Gary O'Neil's volleyed cross was tapped in by Tuncay.
Boro then kept Pompey at bay with ease, while launching breaks of their own.
Portsmouth hustled and harried after the interval and almost levelled but Kanu hit the post from five yards out.
Middlesbrough went into the match looking for their first points of the Christmas period, having lost their previous two games, while Pompey had picked up only a single point.
The hosts were also hoping to end a sequence of seven-and-a-half hours without a goal at Fratton Park.
Initially it looked as though Portsmouth would boss proceedings as they made the faster start.
But too many passes went astray and anyone in a blue shirt who lined up a shot appeared to have the co-ordination of a Christmas drunk.
Boro stayed cool at the back and invited their opponents on, using Stewart Downing as an effective outlet for fast counter attacks.
And it was the England winger who created Boro's goal when he crossed from the right to find O'Neill at the far post, and the former Pompey player volleyed back across goal for Tuncay to tap in.
The pattern of play for the first half was set and the better chances went Boro's way - an O'Neil free-kick just too high, and a miss-kick from Robert Huth at the near post.
By contrast, Middlesbrough keeper Mark Schwarzer was called into action early on to smother a Niko Kranjcar effort but had time to compile his list of New Year's resolutions untroubled thereafter.
Just before half-time, Papa Bouba Diop might have changed things but skied a shot over the bar from eight yards.
Portsmouth tried to quicken the pace in the second half and had some success in that regard but still struggled to find any penetration.
And when they did create an opening on 65 minutes, Kanu wasted it.
Substitute Matthew Taylor, who had replaced Hermann Hreidarsson at half-time, swung in a free-kick from the right only for Kanu to slam the ball against the post from close-range.
Boro might have made Pompey pay almost immediately but Dong-Gook Lee failed to convert two chances in quick succession.
There was no shortage of commitment from either side - demonstrated by one or two ugly challenges - but there was a distinct lack of real quality.
And in the end, Pompey suffered the consequences for that as they fell to their first defeat by Boro in a top-flight game at Fratton Park for 54 years.
Portsmouth manager Harry Redknapp:
"Some of my players looked very tired. I expected to beat Middlesbrough but we never really got going.
"I don't know where the next goal is coming from here for us but we just keep battling on.
"We are going to have to do that because when we lose four players to the African Cup next month we'll be down to a bare 14 in the squad."
Middlesbrough boss Gareth Southgate:
"We were down to just 16 fit players but they were resilient and had the desire and will to win.
"People probably looked at us and thought we'd be rolled over at a place like Portsmouth but I was delighted by the attitude we showed.
"We showed all the qualities we didn't show in the defeats against West Ham and Birmingham and it just shows you what you can achieve."
Portsmouth: James, Lauren, Campbell, Distin, Hreidarsson (Taylor 46), Utaka (Nugent 62), Diop, Hughes (Muntari 72), Kranjcar, Mwaruwari, Kanu.
Subs Not Used: Begovic, Traore.
Middlesbrough: Schwarzer, Young, Woodgate, Huth, Pogatetz, O'Neil (Johnson 86), Shawky, Arca, Downing, Sanli, Lee (Hutchinson 75).
Subs Not Used: Turnbull, Riggott, Grounds.
Booked: O'Neil, Woodgate, Young, Arca.
Goals: Sanli 20.
Ref: Alan Wiley (Staffordshire).
BBC Sport Player Rater man of the match: Tuncay 8.36 (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.