Alvaro Negredo struck twice and Sergio Aguero was also on target for Manuel Pellegrini’s men as they set up a Wembley meeting with either Manchester United or Sunderland on March 2.
The Hammers made the worst possible start as more shambolic defending allowed Gael Clichy to advance unchecked, he fed Marcos Lopes on the left and the youngster’s cross was converted by an unmarked Negredo from just eight yards.
And it was 2-0 in the 24th minute thanks to more excellent work from 18-year-old Lopes, who wriggled away from three West Ham players before flicking the ball through to Aguero, with the Argentina striker leaving Razvan Rat and Matt Taylor on the floor before flicking beyond Jaaskelainen.
Negredo added a third just before the hour mark, showing both power and precision in equal measure, surging into the box and holding off Roger Johnson and Taylor and then beating Jaaskelainen with a deft touch from close range and a tight angle.
Allardyce will also be concerned by the fact that Mohamed Diame and Joe Cole suffered injuries, with the former stretchered off in the closing moments.
With 175 seconds gone Lopes was given a couple of yards of space down the left. He whipped in a pin-point cross for an unmarked Negredo who headed home.
Nolan thought he had equalised, but unlike Negredo he timed his run poorly and his header was ruled out for offside.
Aguero made Johnson, who has spent most of the season on loan at Sheffield Wednesday, look a fool, nipping the ball past the defender, but luckily for West Ham the shot from City’s Argentinian flew a yard wide.
Aguero and Negredo were combining superbly. Aided by the quicksilver Lopes and Jesus Navas, City were a pleasure to watch going forward. West Ham were making it easy for the visitors too, though.
Taylor dallied on the ball and City piled bodies forward on the break. Jaaskelainen just got a hand to Aguero’s shot and the hosts breathed again.
Andy Carroll, making his first start of the season, woke from his slumber to power a header high and wide in a rare West Ham attack.
The inevitable second came with 24 minutes on the clock and once again City had Lopes to thank.
The 18-year-old galloped forward from the halfway line and slipped past a trio of West Ham players before finding Aguero via the boot of James Tomkins.
A comedy of errors played out as Rat, and then Taylor slipped, allowing Aguero time to chip the ball past an onrushing Jaaskelainen.
Aguero was tearing around the pitch straight after as he attempted to find a third. His took his exuberance too far on one occasion though, stamping on Tomkins to earn a booking that could have easily been a red.
Carroll tried to belt the ball at goal in frustration at being flagged offside but it ended near the corner flag. The England striker’s night came to a close at half-time when he was replaced by Carlton Cole.
West Ham were short on luck as well as quality. Joe Cole hobbled off with an injury. Stewart Downing came on in his place. Joleon Lescott then took down Ravel Morrison in the box, but referee Chris Foy waved play on.
West Ham offered little more going forward and they found themselves 3-0 down just before the hour.
Negredo held off the challenges of three West Ham players before chipping the ball over Jaaskelainen from the tightest of angles.
Pellegrini kept Negredo on as he replaced Aguero with Stevan Jovetic and the move looked to backfire as the Spaniard tumbled over Tomkins’ challenge.
Foy took no action but Negredo returned to the pitch shortly after some treatment on the sidelines.
There was no such luck for Allardyce though, who saw Diame carried off on a stretcher shortly before the final whistle, which was greeted by loud booing.