In the 27th minute, Vlahovic broke the deadlock for Juve, receiving a pass inside the box from Federico Chiesa and calmly slotting it into the bottom corner.
Six minutes later, Inter equalised as Martinez made an intelligent run into the box, took control of a cross from Marcus Thuram, and fired an angled shot in at the near post.
The result keeps Inter at the top on 32 points and Juve second on 30 points after 13 matches each.
"It's very difficult to score against Juve, especially when they have taken the lead," Inter manager Simone Inzaghi told DAZN.
"The team remained concentrated, we scored a great equaliser. We didn't create a great deal in the second half, but against a strong Juve side who are deservedly second in the table, we will take the point."
Chiesa had the first chance of the match for Juve when he received the ball just inside the box only to fire his effort over the bar, and Inter's Hakan Calhanoglu then narrowly missed the target with a bold long-range strike.
The two sides probed in vain for vulnerabilities in their opponent's defences at the start of the second half, with play becoming increasingly fractious as they each struggled to break through the other's lines.
The stands erupted in boos when Inter sent on Juan Cuadrado, who joined the club in June after eight years at Juve, late on in the match.
"We conceded a very naive goal, especially as it was only a few minutes after our opener. Inter have already shown that when they have spaces to run into, it makes it easier for them," Juve manager Massimiliano Allegri said.
"Fortune favours the brave, so we had to be solid and aggressive. We are finding a better sense of balance than in the early games of the season.
"We must continue to improve, and a solid defence is very important."