Victory saw Harry Redknapp's men go four points clear of fourth-placed capital foes Chelsea and five in front of north London rivals Arsenal, who could only manage a 1-1 draw at home to Wolves earlier yesterday.
This win also saw Tottenham close to within seven points of the top two of City and United, and with a game in hand as well.
Bale broke the deadlock at Carrow Road in the 55th minute, although his opener owed much to Emmanuel Adebayor.
The Togo striker controlled a pass with his back to goal and shielded the ball from several defenders before teeing up Bale who smashed a shot past Canaries goalkeeper John Ruddy.
Twelve minutes later Wales winger Bale doubled Spurs' lead with a goal that was all his own work.
He received the ball in the centre circle and then accelerated away through the heart of Norwich's defence with a brilliant turn of speed.
As Zak Whitbread tried forlornly to get across, Bale bore down on goal and then deftly chipped the advancing Ruddy to make it 2-0.
'You have to keep grinding out chances and hope one will come for you and it did,' Bale told Sky Sports.
'We learned a lot from being in the Champions League and I'm improving all the time,' he added.
Delighted Spurs manager Redknapp said: 'Gareth is an amazing player. I said to him 'go and play where you want to play'. He gets in holes and when he turns and runs at you, he's unplayable.'
Redknapp, asked what it would take for him to say Spurs were title contenders, said: 'If there's one game to go and we're still in with a chance. It's not impossible but Manchester City and Manchester United look so strong.
'We want to get back in the Champions League, if that's possible.'
Arsenal had to be content with a point despite Wolves being down to 10 men for the final 15 minutes at the Emirates Stadium.
Gervinho gave Arsenal an eighth-minute lead when, following a superb through ball from Israeli playmaker Yossi Benayoun, the forward rounded Wolves keeper Wayne Hennessey.
But Wolves were level shortly before half-time when a corner came out to Stephen Hunt, whose shot took a deflection before Steven Fletcher reacted quickly to head in an equaliser.
Wolves were a man down in the 75th minute when Nenad Milijas was sent off by referee Stuart Atwell for diving in on Mikel Arteta.
Arsenal pressed for a winner, with Robin van Persie twice denied by Hennessey before the Welshman saved well from Thomas Vermaelen as Wolves held out.
'It is very frustrating,' said Arsenal manager Arsene Wenger. 'You won't play many draws like that in a season but sometimes it can happen.'
Elsewhere, promoted clubs Swansea and QPR shared the points in a 1-1 draw. Welsh side Swansea took a 14th-minute lead through striker Danny Graham but QPR levelled at the Liberty Stadium shortly before the hour mark.
Rangers keeper Paddy Kenny's goal-kick was headed backward by Swansea midfielder Leon Britton and that gave QPR forward Jamie Mackie the chance to shoot between Michel Vorm's legs.