The 25-year-old joins the Easter Road club on a three-year contract after leaving Bolton Wanderers.
He spent 17 years with the English Championship club, playing 182 games.
"Josh is an all-round midfielder with the athleticism to support attacks and play on the front foot, helping us win the ball back," said Heckingbottom.
"He was worth the wait and I'm glad we were patient in order to land him."
Vela, who made his Bolton debut in the Premier League as a teenager, played in 17 Championship matches last season as the financially-stricken club were relegated.
"Hibernian were persistent when I had offers from other teams and that played a big part in me coming here," he said.
"Every player wants to feel wanted and I was really impressed with everything that I heard, then what I saw when I came for a look myself.
"The manager went into real detail about he wants from me - as a person and tactically - and I feel like I can help him achieve what he wants to do with the club."
Vela, who also has experience at right-back, is Hibs' seventh signing of the summer.