Tomas Soucek equalised for the second time from a corner and although they had to settle for a point, there were plenty of positives for Graham Pottter ahead of a busy period.
Albion settled quickly into their stride, with Welbeck firing over from an angle and Maupay seeing a shot deflected for a corner during a bright start.
West Ham gradually worked their way into the game, and Jarrod Bowen had a decent chance in the 15th minute when Aaron Cresswell found him in space in the box, but he volleyed well over.
That was the hosts’ best moment of the first half as the Seagulls bossed possession and were first to a lot of second balls, with Solly March at the heart of things. Welbeck was penalised for a foul as he attacked March’s testing cross before the wing-back picked out Adam Webster with a corner, which the defender met with a firm header straight at Lukas Fabianski.
Burn then found March in space down the left and he tried to beat Fabianski at his near post but found the side netting instead. Leandro Trossard then picked out Welbeck with a clever pass but Fabianski was out smartly to save at his feet.
March then threatened again, ghosting past Vladimir Coufal before forcing Fabianski to save diving to his right and it was no surprise when he played a part in the opening goal just before half time.
Found by Burn, he waited for the defender to overlap and Burn picked out Maupay on the edge of the six-yard box. His first shot was blocked but the striker pounced on the loose ball when Declan Rice miscued his clearance and steered it inside the left-hand post.
After a poor first half, it was no surprise to see the hosts upping the tempo having also made a double change. Rice found Sebastien Haller with a decent cross and his header forced Rob Sanchez to make his first save of the afternoon.
But the Albion keeper could do little to prevent the Hammers from levelling on the hour. The visitors didn’t deal with Andriy Yarmolenko’s right-wing cross and when Manuel Lanzini laid it into his path, Johnson fired it into the top corner.
On the balance of play, it was scarcely deserved and for the first time the Seagulls were under pressure as West Ham began to get in crosses from both flanks, a familiar source of goals for them this season.
But with 20 minutes to go, Albion went back in front. Trossard won a corner, found March and when Soucek tried to head away in the six-yard box he found Dunk who steered it home with his right foot.
Soucek made amends with eight minutes to go when the hosts levelled for the second time – powering home a close range header after Dunk failed to head away Cresswell’s corner from the right as both teams were made to settle for a point.