James gave Wales the perfect start to the championship when he opened the scoring as early as the fifth minute.
The Gloucester flanker charged from close range over to stun a partisan home crowd inside the Avcjala Stadium with an early try that was converted by Scarlets fly-half Leggatt-Jones.
Cardiff wing Cummings then skipped over with an impressive finish down the left to extend Wales’ lead in the 16th minute.
Powerful number eight Mikheil Shioshvili was highlighted as one of the tournament’s brightest prospects, not least after playing in a key role in Georgia’s 23-21 victory over New Zealand ahead of this tournament.
And so it proved when the Toulon back-row barged over from a driving line-out on 20 minutes and then bulldozed over just before half-time to cut the Welsh to 15-12 at the break.
Bowen was yellow-carded for a high tackle in the opening minutes of the second half and the hosts hit the front for the first time.
They kicked to the corner for tight-head prop Gabriel Razmadze to drive over but Wales responded despite being down to 14 men.
Full-back Lewis Edwards had a try chalked off after losing control of the ball when going over the line but Leggatt-Jones made no mistake in the 52nd minute, darting over after centre Steffan Emmanuel had gone close.
The fly-half converted and then added a penalty to make it 25-17 in the final quarter.
Wales: L Edwards; R Cummings, O Darwin-Lewis, S Emanuel (co-capt), T Bowen; C Leggatt-Jones, S Davies; G Tuckley, T Howe, J Pritchard, W Evans, T Cottle, D Gwynne (co-capt), C James, E Minto.
Replacements: O Thomas, D James, Y Cook, O Williams, S Morgan, C Gealy, L Lucas, D Scott.