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'):
April 22, 2018
It would become 3-1 three minutes later though through Alexandre Lacazette. The Frenchman picked the ball up on the right-hand side of the box, thumping an effort beyond Hart at the near post thanks to a slight deflection off Aaron Cresswell.
Lacazette (85'):
April 22, 2018
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'):
April 22, 2018