Adrien Rabiot scored a brace either side of Vlahovic's cool second-half finish as Juventus triumphed 3-1 to boost its hopes of progressing from Group H, but Di Maria was the star of the show.
At 34 years and 233 days old, Di Maria became the oldest player to record a hat-trick of assists in the Champions League since data began in 2003-2004, as well as the first player to do so for an Italian club.
Since Di Maria made his first Champions League appearance in the 2007-2008 season, meanwhile, only Lionel Messi can match his tally of 35 assists in the competition.
Asked about Di Maria's influence, Vlahovic said: "I don't know what to say. I am honoured to have the opportunity to play with him, to see him every day, to learn from him. I would like to make the most of my moments with him on the pitch."
Vlahovic was pleased to see Juventus end a three-match losing streak in the Champions League, but acknowledged there is room for improvement in his own performance.
"I could do better and do more," he said. "I had to make better use of the assists of my team-mates, but we won. We have to improve and work with our heads down in silence.
"Attackers live on goals. I am always focused on the team, then the goals come as a consequence.
"If I think I have to do something, I just put weight on myself and I don't do things with clarity. I have to be more calm and focused."
Meanwhile, under-pressure Bianconeri manager Massimiliano Allegri praised Di Maria's display, but was concerned by his team's lack of control in the final 15 minutes.
"Di Maria is good, the important thing is to get the ball to him, the first and second assists were of excellent craftsmanship," he said. "The boys immediately realised that you cannot not play for a quarter of an hour.
"The games must be closed earlier, this does not mean to stop playing, but to play less and be more bad.
"We were too light, then the games are about psychology, and we risked 2-2. This doesn't have to happen.
"The defensive phase is a mental issue. Everything seemed easy, we gave up as a team. We didn't foul [ahead of] their goal, that's not good. Goal difference will also be important."
Looking ahead to next week's return fixture in Israel, two-goal hero Rabiot said: "We will have to be concentrated, but with this attitude, I am convinced that we will win. We must continue like this.
"I'm happy to have scored, but it's more important to win, and tonight we did it well. We played well - a little less in the second half - but we're on the right path."