Bale continued his recent resurgence by marking only his third league start of the season with the opening goal after only two minutes, prodding in from close range.
Bale then claimed his second of the game with a clinical finish after 55 minutes as Tottenham returned to form in the league having lost five of their previous six games.
The victory left Tottenham in eighth spot with 39 points from 25 games and revived their hopes of finishing in the top four.
Bale's lack of Premier League game time has been something of an anti-climax after he returned in October on loan from Real Madrid having left in 2013.
But the four-time Champions League winner is finally beginning to show the kind of form that persuaded the Spanish giants to sign him.
He has now scored four times in his last four appearances in all competitions and although he did not register in the loss by West Ham United last week he was eye-catching as a second-half substitute.
Looking fit and sharp, Bale linked up superbly with Kane and Son Heung-Min as manager Jose Mourinho unleashed the trio together, albeit against a Burnley side who were there for the taking.
"It's not just about scoring. It's about our all-round play and helping the team. I'm delighted with the goals and the assist and whatever I can do for the team," Bale said postmatch.
"I've been happy all season. In the dressing room I feel comfortable and I'm having fun. I'm happy and I'm sure it's showing on the pitch."
A flowing Tottenham move after two minutes ended with Bale ghosting on to Son's astute pass to touch past Nick Pope.
Bale then spotted the run of Kane with a sublime diagonal ball and the England captain's shot deflected past Pope for his 14th Premier League goal of the season.
It was all too easy for Tottenham and Moura made it 3-0 when Sergio Reguilon's ball into the box was only half cleared and the Brazilian fired past Pope.
Tottenham's tempo dropped a little after the break but Bale capped a fine display with a cultured left-foot finish into the bottom corner from Son's feed.
It was Tottenham's biggest home league win since they thrashed Burnley 5-0 last season.
Burnley are 15th with 28 points.