Albion produced a fine first-half performance in front of their new head coach Roy Hodgson, who was watching from the stands, scoring twice in the opening eight minutes through Graham Dorrans and Jerome Thomas before Winston Reid's own goal put them 3-0 ahead just after the half-hour mark.
But things could not have been more different after the break as a Ba double - on his first start for the Hammers - and a header from Carlton Cole in between ensured the match finished all-square.
The Baggies, overseen by caretaker boss Michael Appleton, served up a treat for Hodgson in the first half but the former Liverpool manager, who was announced as Roberto Di Matteo's successor yesterday and takes up the reins on Monday, will be deeply concerned by the capitulation which followed.
West Brom remain above the relegation zone on goal difference alone, while West Ham are two points below them, having moved off the bottom up to 19th.
The hosts appeared eager to impress Hodgson from the off and they made the best possible start as Dorrans put them in front after only three minutes.
The Scotland midfielder, who had not scored all season, opened his account in spectacular fashion, unleashing a fierce drive from outside the box that flew into the top corner.
Two minutes later Albion almost doubled their lead as Robert Green palmed Peter Odemwingie's cross to Chris Brunt, whose shot was on target but cannoned away off his team-mate Marc-Antoine Fortune.
That was a near miss, but West Brom were looking rampant, and within moments, they had made it 2-0.
Fortune made amends by providing the assist, laying the ball to Thomas who then coolly passed it across Green and into the net.
The momentum showed no sign of letting up as Thomas muscled past two defenders, rounded Green and went down under the attentions of Reid, but referee Lee Mason waved away appeals for a penalty.
Youssouf Mulumbu then had to be withdrawn for James Morrison, but West Brom remained on the front foot, with Odemwingie sending a free header from Dorrans' corner into Green's arms.
West Ham thought they had pulled one back when Cole was slipped in and beat Boaz Myhill, but the effort was disallowed for offside.
Reid then survived another scare at the other end when his apparent handball in the box went unpunished.
The New Zealand defender was having a torrid afternoon and it got worse moments later as Albion extended their advantage.
Dorrans sent a free-kick into the area and it deflected in off a hapless Reid to put the Baggies three up.
Hammers captain Scott Parker tried to respond but Gonzalo Jara did enough to put him off as he shaped to shoot and then Myhill tipped Ba's strike on to the post, Morrison clearing the danger.
But West Brom were soon attacking again, with Dorrans bringing a great save out of Green and Thomas shooting wide from distance.
West Ham saw another attempt hit the woodwork in first-half stoppage time as Gary O'Neil cracked a shot against the bar, but the rebound went unconverted, as did a Manuel da Costa header soon after.
The visitors needed something quick at the start of the second period and found it five minutes in thanks to Ba, who chested down Mark Noble's lofted pass and tucked the ball past Myhill.
The goal seemed to galvanise the visitors and Myhill had to get down to stop Luis Boa Morte's header from a corner.
Hammers boss Avram Grant then threw on Frederic Piquionne for Boa Morte and the striker's impact was almost immediate as he nodded Noble's free-kick back across goal for Cole to head in and make it 3-2.
Suddenly, with more than half an hour to go, it was game on and Piquionne nearly drew things level with a header that bounced off the bar.
West Brom managed to switch play and Odemwingie was denied from the edge of the box by another fine Green save.
But West Ham were in the ascendancy and went close again when Ba's shot was blocked by Nicky Shorey.
The Senegal forward then sent a shot into the side-netting, and with seven minutes of normal time remaining, he scored the third West Ham goal which had by that point seemed almost inevitable.
Noble was the provider once more, delivering a cross that was not dealt with and fell to Ba, who volleyed against the turf and in.
Odemwingie and Piquionne had chances to seal victory in the dying moments, but it was honours even come the final whistle.