The Hammers suffered the worst FA Cup home defeat in their history on Friday after Yaya Toure set the ball rolling in the 32nd minute when he lashed home a disputed penalty after Angelo Ogbonna tangled with Pablo Zabaleta.
West Ham could have equalised had Sofiane Feghouli managed to find an unguarded net rather than slide the ball across goal.
But instead they capitulated as Havard Nordtveit's own goal and Silva's tap-in made it game over before half-time.
Aguero chipped in with the fourth five minutes after the interval and John Stones headed a late fifth to cap a chastening evening for the Hammers.
Manager Slaven Bilic called on his side to quickly put the result behind them, with a crucial league game against Crystal Palace next up.
"We put a strong line-up out, a team we thought would have a really good chance and go through to the next round," he said.
"Unfortunately it wasn't good but now it's time for the team to show character.
"We've had a few defeats like that, let's say after the (5-1) Arsenal defeat we showed character and took 10 points from our next four games. Now is the time to do it again."