It was a lively start to the clash as Craig Goodwin put the visitors ahead in the third minute with a tap-in at the far post after latching on to Tarek Elrich's ball.
Goodwin was arguably Adelaide's best player in the first half as he orchestrated the attacks, but the visitors suffered another blow when Krishna snatched possession and fed the ball to Roly Bonevacia, who netted his side's second goal of the match.
Despite Adelaide relinquishing their lead, Goodwin tried to make things happen with two good attempts - one was a header, which ended off target, while another shot curled wide of the far corner.
Wellington had their chances too as Krishna pulled off a back heel to find Alex Rodriguez, but the midfielder fluffed his shot from close range as the ball soared over the bar.
After the break, Moss was called upon to make another save as Goodwin's free kick looked on course for goal, but Adelaide's misery continued when Bonevacia netted strike to pull Wellington clear at 3-1.
Jeffrey Sarong could have got on the scoresheet if his effort had not bounced off the post, but the home side eventually nabbed a fourth goal when Matthew Ridenton snuck the ball in at the far post around 20 minutes from time.
Adelaide tried to find a way back into the game, and they managed to claim a goal from Pablo Sanchez in the 89th minute, but it was only a consolation for the visitors.