First-half goals from Philip Billing and Dominic Solanke – both set up by Ryan Christie – put the Cherries in control.
They were unable to find an equaliser, however, as the Cherries leapfrogged West Bromwich Albion to move top.
Bournemouth went ahead when a neat move culminated in Christie lifting the ball towards Billing, who volleyed home.
Solanke headed in Christie’s left-wing cross on the half-hour mark to double the lead and take his goal tally for the campaign to seven.
The deficit was reduced when Luke Berry’s cross towards Luton defender Reece Burke cannoned into the net off the unfortunate Kelly.
Bournemouth’s win was their fourth in a row and they lead the second-placed Baggies by two points.
They have now scored 11 goals in five home league matches this season.
Luton, meanwhile, were left to rue two misses by Elijah Adebayo, who unable to take an early chance when the game was goalless and his team were on top.
The striker missed another opportunity in the closing stages as the hosts clung on.
Bournemouth manager Scott Parker told BBC Radio Solent:
“We played very well in the first half and the game then runs away from us a little bit.
“That’s part and parcel of football matches. Are they ever going to be, from nought to 90, exactly how you want? I’ve never had that as a coach.
“What is important is that we win the game and we dig in. In the second half, as much as they put pressure on us, we defended it and managed the game well.
“We’d like it to be more comfortable, but you’ve got another 11 men who are trying to score a goal. I thought we did well.
“They had nothing to lose and when you have nothing to lose you’re a bit more gung ho.
“Winning is a habit and at this moment we’re doing good on that.”
Luton boss Nathan Jones:
“We had the best chances of the game, really. We put it over from two or three yards and you can’t do that at this level.
“Second half I thought we were excellent, really front-footed, but we’ve had to chase too many games lately.
“We’re in the top three in chances created but we’re just not converting them at the moment.”