Now Burrows, with 920 games in the NHL, including 78 in the Stanley Cup Playoffs, is on the other side trying to use everything he learned from Vigneault against his former coach in the Eastern Conference Second Round.
The Senators lead the Rangers 2-0 in the best-of-7 series thanks in part to Burrows' play, diving and chipping the puck up the ice, to set up Jean-Gabriel Pageau's double-overtime winner Saturday.
Game 3 is at Madison Square Garden on Tuesday (7 p.m. ET; NBCSN, CBC, TVA Sports).
"He's an example I think any coach in the League could use about a guy figuring it out and making it, but I never had to push him," Vigneault said before Game 2. "He was pushing himself."
It's obvious how much affection Vigneault still has for Burrows. When asked about him, Vigneault seemed excited at the chance to retell Burrows' story, the Cliffs Notes version, of perseverance.
He talked about how at 19 years old Burrows went to play with Shawinigan in the Quebec Major Junior Hockey League as an overage player after completing a prolific ball-hockey career. He talked about his two-plus seasons in the ECHL, playing in South Carolina and Louisiana. He mentioned his time with Manitoba of the American Hockey League.
Burrows, undrafted, didn't make his NHL debut until he was 24 years old.
"I think he's a real good story of a player that figured it out," Vigneault said. "He made it on sheer will and determination.
"Probably everybody was telling him he couldn't make it. He did make it. He's a great story of perseverance."
\[RELATED: Complete Senators-Rangers series coverage\]
Burrows said he wouldn't have been able to persevere without Vigneault's guidance.
"He was always good for me," Burrows said. "He made me become a better player, understand the game better, how to play the right way, the high-percentage way.
"I was just a young guy who thought maybe the League was easier than what I expected it to be. I came in, my first year I had a few goals and I might have said this league isn't as hard as people said it is, but when you go after a tough year, you really realize this league is no joke. You have to prepare the right way. You have to train the right way. You have to put all your effort in. Just the trust, the communication, the ups and downs, he knew."
Vigneault still knows the damage Burrows can cause, that he'll continue to be in the face of Rangers players and goalie Henrik Lundqvist as the series shifts to New York, that he won't hesitate to speak his mind on the ice.
Vigneault knows how valuable Burrows is to a team in the playoffs.
Burrows had 32 points in 67 postseason games under Vigneault with the Canucks. He helped them reach the Stanley Cup Final in 2011, when they lost in Game 7 to the Boston Bruins.
"We had a lot of good years, so obviously he's meant a lot to me," Burrows said. "We had some good success. We won [five] division titles in a row, a couple Presidents' Trophies, the Stanley Cup Final. But ultimately our goal was to win the Stanley Cup and we weren't able to accomplish that. Hopefully, I'll be able to get it done before he does."
He'll probably call Vigneault to thank him if he does.