Chelsea defender Marcos Alonso scored twice to rescue a point in a thrilling 2-2 draw against Bournemouth at the Vitality Stadium on Saturday.
Chelsea stay in fourth position on 45 points, while Bournemouth are still hovering above the drop zone on 27 points, two ahead of 17th-placed Aston Villa.
Billing had another opportunity moments later but his strike hit the side netting with Callum Wilson in a much better position.
The visitors came close to scoring a second right after the break but Giroud's close range shot went wide and Jack Stacey cleared Alonso's strike off the line.
Bournemouth made them rue those missed chances and scored two goals in three minutes. Lerma equalised with a powerful from a corner and King got on the end of Stacey's cross to tap home.
Jolted into all-out attack, Chelsea substitute Michy Batshuayi had a goal ruled out for offside moments after coming on, but they continued to bombard the Bournemouth box with set-pieces and crosses.
Chelsea grabbed an equailser with five minutes remaining after Alonso reacted first to Aaron Ramsdale's save to head the ball in.
Despite the late equaliser, Chelsea manager Frank Lampard was disappointed not to get more out of a game that his side dominated for long periods.
"We should win, we had many chances. We had a bad 10-minute spell where they scored and that I don't like. I'm pleased with the character to get a draw but we should win the game," he told the BBC.
"People talk about the defence, but if you are 1-0 up and leave teams in the game then one set-piece can change the game. We should score more goals and then we don't have that problem."
Bournemouth boss Eddie Howe had mixed feelings after coming back from a goal down and then taking the lead only to let Chelsea back into the game at the end.
"It's a glass half-full day, we were 1-0 down and it's hard to come back against these types of teams. We showed great spirit, we just couldn't carry on great starts to both halves," he said.
"We need to get points. We have lost some narrow games, so hopefully this is a turning point."