The visitors were on top in the opening exchanges, but Roy Keane's side took a stranglehold midway through the first-half that they did not surrender.
Matthews sent a powerful header into his own net from Carlos Edwards' cross in the 62nd minute at Portman Road.
And striker Scotland produced a low shot finish following a break-out.
After Keane's side got into their stride, the Bluebirds, without injured skipper Craig Bellamy, were regularly under pressure, particularly whenever Connor Wickham was anywhere near the ball.
The 17-year-old was making his first appearance of the season and his battle with Cardiff defender Gabor Gyepes was one of the game's combative highlights.
The Ipswich man was yellow-carded for a late, clumsy first-half challenge on Lee Naylor and he risked a second after the break by bringing Gyepes down.
But as Wickham gradually got the better of his man, so the hosts took an increasing grip on the game.
Gareth McCauley's flicked on header bounced harmlessly past David Marshall's far post and as Ipswich's pressure increased, Wickham created a great chance, but was unable to capitalise on the rebound.
Marshall pulled off a brilliant save after Wickham proved too strong for Gyepes and Mark Hudson, the keeper pushing the shot off the floor and over the bar.
But despite all of Ipswich's Wickham-inspired efforts and Cardiff's resolute defence, the opening goal was stylish if unfortunate for 18-year-old Matthews.
Edwards sent what seemed a harmless cross into the Cardiff box and Matthews met it with a header timed to perfection for the own goal.
Peter Whittingham found his path blocked as Cardiff upped their efforts and Jay Bothroyd was also denied before ex-Swansea City player Scotland finished off a break-out with a well-taken, low drive from close range.
Marton Fulop got a hand to a Whittingham header as Cardiff sought an immediate response and Chris Burke fired over the top in time added on.
Source: BBC Sport
Other reports:
The Telegraph
Cardiff Official Website
Wales On Sunday