The 'Hammers' had done well to restore parity in the second half but they would see victory slip away as Arsenal hit back with three quick goals to secure all three points at the Emirates.
Marko Arauntovic had restored parity in the 64th minute after Nacho Monreal's opener just after the break.
However, the scores would not remain level for long as Arsenal struck again in the 82nd minute, although they were gifted some luck by the visitors.
Ramsey whipped a ball in the box from the left which Rice inexplicably ducked underneath, leaving his goalkeeper wrong-footed and unable to do anything as it sailed into the far corner.
Ramsey (82'):
Lacazette would strike again four minutes later to make it 4-1. Ramsey danced his way to the byline down the left before pulling the ball back for his team-mate who let the ball run across him before firing it low past Hart.
Lacazette (89'):