Neither side were able to find a winner on a cold night in the capital.
"We didn't play particularly well with the ball, we didn't pass it well," West Ham manager David Moyes told BBC Match of the Day.
"Anyway, we got something out of it and we'll take that. We'll hopefully improve."
The Hammers, without a win in their past four games, stay sixth in the Premier League table, while Bournemouth move 12th, eight points clear of the relegation zone.
Phillips gets rare start at London Stadium
Manchester City loanee Phillips has struggled for game time since joining Pep Guardiola's City from Leeds for £45m in July 2022.
On Friday night he made only his third Premier League start in the past two seasons, and with only his second-ever touch in a West Ham shirt, prodded Kurt Zouma's poor pass straight to opponent Solanke who finished from close range.
West Ham's equaliser came after the Video Assistant Referee (VAR) told referee Tim Robinson to look at his pitchside monitor, and the official reversed his no-penalty call.
Ward-Prowse levelled from the spot, but the hosts were unable to create any further clear-cut chances and were even booed off by a smaller number of supporters at full-time.
The Hammers have now failed to win any of their opening five games in a calendar year for the first time since 2010, though they are still in a European spot in the Premier League table.
As for Phillips, he did tighten up after the break and came off for Danny Ings to a warm reception in the 68th minute.
"I know the kind of player Phillips is and the quality he has," said team-mate Kudus to Premier League Productions. "When you come to a different team and system it takes time to get used to everything.
"I believe in him so much and I have no doubts about him. Next game he will be better."
Bournemouth will rue missed chances
Bournemouth will be disappointed not to leave London with all three points having created chances at will in the first period.
After the match manager Andoni Iraola told BBC Match of the Day: "I think if someone had to win today it was Bournemouth."
Marcos Senesi also volleyed over the bar while Semenyo had another effort tipped over by Areola on the stroke of half-time.
Even after West Ham's leveller the Cherries had the better chances, Kelly having an effort saved from a tight angle and Ryan Christie dragging a shot wide.
"We had the most clear chances," added Iraola. "We probably weren't clinical enough to score a second, go 2-0 up and win the game.
"OK, we get just one point today but we have to value it. We were closer to the three points than losing. We have to continue pushing."
Bournemouth enjoyed success in the FA Cup in January, but are still waiting for their first Premier League win since 26 December.