Maynard, a product of the famous Crewe academy, opened the scoring for the Hammers with a sweet volley from inside the box and in front of the travelling support.
Modibo Maiga marked his full debut by doubling the lead from close range after the interval as West Ham eased into the third round.
Maynard first joined Crewe as a seven-year-old and scored 35 goals in 67 appearances for the club before moving to Bristol City in 2008, for £2.25million, and then to West Ham in January.
The 25-year-old, who has been linked with a move away from Upton Park, took his chance to impress tonight as a youthful West Ham side flooded forward.
Manager Sam Allardyce made 10 changes from Saturday's 3-0 Premier League defeat to Swansea, with captain Kevin Nolan the only player retained in the side.
New arrivals Maiga, Alou Diarra and Stephen Henderson made their first starts for the club while the likes of Maynard were given their first chance of the season to impress.
After Jordan Spence had looped a header over the bar, Maynard cut in from the right flank and attempted a curling left-footed strike which flew wide.
Henderson made a nervy start, punching a ball he should have caught. It deflected off Kelvin Mellor's knee and, thankfully for the West Ham goalkeeper, over the bar.
Crewe conceded five against Brentford at the weekend but they were resilient at Upton Park, keeping West Ham at bay until Maynard's 32nd-minute opener.
The visitors had their moments. Ajay Leitch-Smith tried to tee up Max Clayton on the edge of the West Ham box but they were denied by a timely interception from Matt Taylor.
Byron Moore then nutmegged Taylor and cut into the penalty area before pulling the ball back to Clayton but Joey O'Brien reacted smartly to shut down the danger.
West Ham were dominating but they were not ruthless enough. They began to get sloppy in midfield and Ricardo Vaz Te almost blasted one shot into the upper tier of the Sir Trevor Brooking Stand.
But then came the goal. Maiga and Vaz Te created the opening with a sharp one-touch move and Maynard, with a perfect technique he learned at Crewe, swept the low volley into the bottom corner.
West Ham's gander was up and they won a corner, which fell to Maynard but this time his fierce volley from inside the box was well saved by Alan Main.
The Crewe goalkeeper was a busy man and he needed to produce another stop to keep out Diarra's header and preserve West Ham's 1-0 lead at the interval.
Crewe's Luke Murphy saw a stinging 30-yard shot tipped around the post for a corner and the visitors then had strong claims for a penalty when Harry Bunn went down under a challenge from Diarra.
The Frenchman was desperately trying to rescue the situation after West Ham had lost possession on the edge of their own box and referee Fred Graham waved away Crewe's appeals.
West Ham effectively wrapped up the win when Nolan received a free-kick from Taylor, forced Main to parry his shot and Maiga pounced to score from close range.
The only issue that will furrow Allardyce's brow after the victory was how Maiga only scored one.
Shortly after the goal he met Taylor's deep cross with a stretched volley that went over the bar from just six yards out.
West Ham were in their stride now and Main was bombarded with shots as he denied Vaz Te's fierce drive and then two more long-distance efforts from Maiga.
In between time, Henderson did well to keep his feet and deny Leitch-Smith as the Crewe striker tried to round the Hammers keeper.
West Ham finished on the front foot with Maiga directing a header just wide and substitute Robert Hall just missing with an angled shot.As West Ham accepted the victory applause, Maynard made a point of acknowledging the Crewe supporters he had helped to send home disappointed.
|Team Name||West Ham United||Crewe Alexandra|
|Shots On Target||10||4|
|Shots Off Target||11||3|