Nuno Espirito Santo's side had been held to draws by London rivals Millwall and QPR in their previous two matches.
But the third-placed Hammers - relegated from the Premier League last season - ended that frustrating spell in emphatic fashion at Sincil Bank.
Lincoln duo Tom Hamer and Conor McGrandles were sent off in the third and 23rd minutes respectively.
Konstantinos Mavropanos scored West Ham's opener in the 45th minute before Edvin Austbo, Jarrod Bowen, Pablo and Tomas Soucek, who bagged a brace, all netted for the east Londoners, who also beat Wrexham 6-0 earlier this season.
West Ham are four points behind leaders Swansea, who won 1-0 at Millwall on Tuesday, ahead of their trip to face the table-toppers in Wales on Saturday.
Southampton clinched their second 1-0 win in the space of four days as they handed Norwich a fourth successive league defeat.
The Saints followed up Sunday's victory over arch rivals Portsmouth with their first away league victory this season, with Kuryu Matsuki scoring the decisive goal early in the second half at Carrow Road.
Tonda Eckert's side are up to seventh place despite starting the campaign with a four-point deduction, while Norwich are languishing in 20th place.
Ebou Adams scored a last-gasp winner as Portsmouth beat 10-man Sheffield United 3-2 at Fratton Park.
Zak Swanson and Adrian Segecic netted for the hosts either side of United midfielder Hamza Choudhury's red card for two bookings.
Japhet Tanganga and Tom Cannon scored to haul United level but Adams came off the bench to bag the 89th-minute goal that gave Portsmouth their third league win this term.
QPR are fifth are thrashing second-bottom Derby 4-1 at Loftus Road.
