It was the first time Hart played at his old club as an opposition player since leaving to join the Clarets in 2018, although he had spent his last two seasons at City out on loan after Pep Guardiola deemed him surplus to requirements.
The former England number one enjoyed a successful spell at City and was a key member of their title winning sides in 2011-12 and 2013-14.
"He was paramount to his team, he made some incredible saves on the line," Kompany told reporters after the Burnley win. "I can only reiterate how much of a legend he is for this club. It can't be under-estimated what an impact he's had on helping to build this place and make it what it's become today.
"You see his talent shine through in a goal where he's used to playing an important part. I want him to do well and he looks like he's happy. That's the main thing."
Hart was a prominent leader in the City dressing room during his time in Manchester and Kompany stressed the importance of his character to the teams he plays for.
The Belgian continued: "You take away what you see on the pitch, his ability to stop goals - Joe is a winner. He's always been a massive personality and voice.
"Having him in your team makes a difference by his presence. That's the side that people have never really gotten to understand. In a team like he's in today, you can see the lads fighting for each other, a togetherness. He's able to lift that team."
Prior to the match, Guardiola admitted that it may have been a mistake to let Hart go the way he did while complimenting the veteran goalkeeper.
Although Hart will have been disappointed with the manor of his side's defeat at the weekend, Burnley sit in a comfortable 13th place in the Premier League table, three points above the relegation zone.
The win for City meanwhile ensured that Guardiola's team stayed ahead of Liverpool on goal difference at the top.