Said Benrahma salvaged a point with a superb late goal at the Amex Stadium following Danny Welbeck's clipped finish, but the result leaves the Hammers five points off the top four with just two games remaining.
Moyes' men had the better of the chances on the south coast, but fell behind during a frantic finale when Welbeck claimed his 50th Premier League goal just six minutes from time.
The east London club have games against West Brom and Southampton to ensure the minimum of a Europa League place does not slip through their grasp, with Tottenham and Everton each just three points behind and in hot pursuit.
Hammers boss Moyes had no complaints about his side's display on Saturday evening but was less enamoured by their cutting edge.
"I thought we played very well for long periods of the game, especially first half," said Moyes.
"I thought Brighton improved in the second half, we didn't do enough in the final third. I thought our decision-making in the final third was poor.
"But overall I thought we played very well tonight. We didn't show enough quality in the pass, quite often it was the decision-making."
Algeria forward Benrahma drilled home his first West Ham goal from just outside the box to swiftly cancel out Welbeck's landmark strike and prevent his side slipping to a fourth defeat in five outings.
Hammers defender Craig Dawson had earlier squandered a golden chance to open the scoring with a free header, while Pablo Fornals wasted three presentable openings in the first half.
Moyes hopes Algeria forward Benrahma can kick on after ending a lengthy drought following his big-money from Sky Bet Championship club Brentford.
"We've been waiting for a goal and we got one tonight," added Moyes.
"We brought him in with that in mind and I am glad he has got that first goal and hopefully he will go on to get a lot more."
Additional reporting by PA.