James Hayter handed Yeovil the lead in the first half and with 16 minutes remaining Lawrence, on loan from Manchester United, doubled the visitors' lead.
A stunning 78th-minute strike from Elliot Grandin gave Blackpool hope but bottom side Yeovil squeezed home.
The home side fashioned the first chance of the game after four minutes as Gary MacKenzie flicked a header on to David Goodwillie who could not quite lift the ball over Yeovil keeper Chris Dunn.
Yeovil came close after five minutes as a long free-kick forward by Jamie McAllister fell to Liam Davis who hit a powerful left foot shot just over the bar.
Yeovil took the lead through Hayter from the penalty spot in the 16th minute after Blackpool defender Tony McMahon was adjudged to have handled in the area from a corner.
Barry Ferguson's side had a golden opportunity to equalise after 24 minutes as MacKenzie met a McMahon free-kick from the right but headed over unmarked from six yards.
Yeovil had the chance to double their advantage after 34 minutes as their continued pressure led to Grant nodding on to Hayter who could not poke past Gilks from 12 yards.
Goodwillie tried to turn momentum Blackpool's way with a strong right-footed strike from 35 yards, which whistled just over the bar.
Just before half-time, the frustration grew among the Bloomfield Road crowd who booed Jack Robinson for another backwards pass and Robinson responded by shouting back at the supporters.
Blackpool, who were booed off at half-time, came close to conceding again after 48 minutes. The ball broke to Grant from a free-kick, who forced Gilks to save his right-foot smash from eight yards.
Goodwillie squandered a fantastic opportunity to equalise after 53 minutes, dribbling through the Yeovil defence before inexplicably firing over when one-on-one with the keeper from 12 yards.
Blackpool pushed forward once more and Neal Bishop should have scored from a right-wing Goodwillie cross but headed wide from six yards.
Yeovil doubled their lead after 74 minutes as Lawrence was slipped in down the middle and struck the ball low left-footed between the legs of Gilks.
Blackpool struck back almost immediately as substitute Grandin's dipping shot from 25 yards crept just under Dunn's bar after 78 minutes.
The hosts laid siege to the Yeovil goal in injury time in search of an equaliser but Andy Keogh's shot from 20 yards whistled over and the Glovers clung on for the win.