Lerma walked early in the second half for two bookings and Villa made a game of it as Mbwana Samatta scored on his Premier League debut – becoming the first Tanzanian to do so in this division.
But Eddie Howe's men leapfrogged their visitors in the table after seeing it out, claiming just their second win since mid-December.
Villa, who were on a high after reaching the Carabao Cup final in midweek, had chances to earn a point, but the result left them sitting perilously above the bottom three.
Dean Smith's men expended a lot of energy – both physical and emotional – in getting past Leicester in midweek and the worry before the game was whether there would be a hangover.
They certainly started sluggishly as Bournemouth took early control of the game and could have been ahead.
Billing tested Pepe Reina's handling early on with a drive from distance, the Spaniard then grabbed Harry Wilson's free-kick at the second attempt before Dan Gosling missed a golden opportunity, blazing over from Ryan Fraser's cross.
Led by the slaloming Jack Grealish, who was dancing past defenders at ease, Villa enjoyed their best spell and an enticing ball by the playmaker missed everyone and grazed the far post.
But it was during this period that Bournemouth took a 37th-minute lead.
The Cherries recycled a free-kick and Simon Francis' cross was knocked down by Gosling into the path of Billing who drilled home from 12 yards.
Howe's men got a crucial second goal on the stroke of half-time to put themselves in firm control.
Fraser's effort from the edge of the area was parried by Reina, but he could not get it away from goal and Ake rammed home the rebound from close range.
It could have been three soon after the restart as Lerma grazed the crossbar with an arrowed effort from distance, but the midfielder's next contribution handed Villa a lifeline.
Having already been booked in the first half, the Colombian was deemed to have body-checked Grealish, though it looked soft, and was shown the red card by referee Anthony Taylor.
Callum Wilson could have put the game to bed but he fizzed an effort just wide from Billing's pass and the hosts could have done with that going in.
It took them a while but Villa eventually came to life and gave themselves hope in the 70th minute.
Grealish came close to a leveller when he bundled an effort wide at the far post, while Davis tested Ramsdale with a low shot and then Tyrone Mings headed a free-kick off target.
Villa were pushing until the final whistle, but Bournemouth held on for an important victory.