Lanzini opened the scoring in the 19th minute, before Chrisitan Benteke rounded off a quiet half with a missed penalty and clattering the post.
Dimitri Payet had two good opportunities to double the lead at the beginning of the second period, but were left with a mountain to climb after Aaron Cresswell's dismissal, but managed to secure the three points.
Pressure was starting to build upon Slaven Bilic's side as they headed to Selhurst Park, and started brightly against Alan Pardew's side.
However it was Palace who had the first chance after 11 minutes, it was poor defending from the Hammers, as Wilfried Zaha was left out on the right. He received the ball back after his initial effort was blocked but his effort was just wide of Adrian's post.
The Hammers took the lead just before the 20-minute mark, it was a good start to Cresswell's return. The full-back put an excellent ball and it proved easy for the unmarked Lanzini as he swept it home.
Hero to a villain
It was a pretty drab remainder to the half after Lanzini's opener, but the half ended with a bang as Palace were awarded a penalty.
The torrential rain was proving an issue for the 22 on the pitch, and Angelo Ogbonna's lunge on Benteke gave Anthony Taylor no option.
The Belgian striker looked confident as he stepped up, but he opened his body which saw him stick it wide of the right-hand post.
Benteke nearly redeemed himself almost instantaneously, as he rose for the header in extra-time but clattered the post.
Looking for a second
Bilic's side came flying out of the traps at the beginning of the second period, and had good opportunities to double their lead.
The goalscorer Lanzini looked to turn provider, as he played a over the top ball to Payet, but Frenchman's heavy touch allowed Steve Mandanda to save well.
The duo were at it again moments later, as Lanzini beat Mandanda to the ball before laying it off for Payet, yet his compatriot in the Palace net did brilliantly to block his effort.
Nerves began to jangle as the clocked ticked down, and the Hammers made it more difficult on themselves as they were reduced to ten men.
Cresswell was booked in the 74th minute, but was heading for a early shower only a minute later, as he brought down Zaha and left his side with a job to do.
Pardew's side pushed for an equaliser, having gained the numerical advantage. Benteke came close with nine minutes to play as he managed to head Jason Puncheon's corner over from six yards.
Their best chance came two minutes into added time, as the Eagles piled into the area with Conor Wickham rising highest to Andros Townsend's cross, but Adrian did brilliantly to tip the header over the crossbar.