The Italian striker struck with a downward header in the 30th minute of a match Juventus dominated, handing Leonardo his second defeat as Inter coach since taking over from Rafa Benitez.
Inter stayed fourth, eight points behind leaders Milan who defeated Parma 4-0 on Saturday, with a game in hand while Juve climbed to sixth.
Lazio recorded their first away league victory since October by defeating lowly Brescia 2-0 to go third.
There were also away wins for Udinese and Fiorentina, their first of the season. Struggling Cagliari, Sampdoria and Catania won at home.
Juventus coach Gigi Del Neri said his players won because they were the hungrier side.
"We wanted it tonight. January was a difficult month - we had some injuries and lost form, but this is a good side and we can go very far still if we keep up the same spirit and desire," he said.
Inter coach Leonardo said his team deserved a draw.
"The first half was difficult, we were slow and couldn't get men forward. The second half was very different, we got more players into the box and had chances.
"If we win all our matches, we will win the league but we take each game at a time and so far we have seven positive results out of nine."
Juve took a deserved lead on the half-hour when Milos Krasic laid the ball back to Frederik Sorensen whose instant cross was headed precisely into the bottom corner by Matri.
After three victories in a week culminating in a spectacular 5-3 win over Roma, Inter seemed strangely out of sorts but Giampaolo Pazzini almost equalised just after the restart, flicking a near-post header that Gianluigi Buffon did well to parry.
Juventus, beaten three times at home this season, should have doubled their lead twice but Matri headed a whisker wide and Giorgio Chiellini also nodded just off target from a corner.
Inter pressed and striker Samuel Eto'o had an effort saved on 82 minutes after an audacious flicked pass from Wesley Sneijder.
The in-form Cameroonian was then guilty of one of the misses of the season with two minutes left after smashing the ball against the crossbar from four yards with the goal gaping.
A minute later Eto'o kicked thin air with another presentable opportunity to sum up a disappointing evening for Inter who play their game in hand at Fiorentina on Wednesday.
Earlier on Sunday, a goal in each half from forwards Alvaro Gonzalez and Libor Kozak earned Lazio a surprisingly straightforward victory over Brescia.
Antonio Di Natale kept pace with Edison Cavani at the top of the scoring charts. After the Uruguayan's double against Roma on Saturday took him to 20, the Italian striker bagged his 17th and 18th strikes of the season in Udinese's 3-0 win at Cesena which kept them fifth.
Strong first-half performances led Cagliari and Sampdoria to c
