REUTERS
The hosts, second in Sky Bet League One, dominated for long spells and fully deserved another shot at the Hammers after a 0-0 draw.
West Ham needed Shrewsbury-born Hart, who began his career with the Shropshire side, to save well from Mat Sadler and Shaun Whalley while Jon Nolan fired over late on.
The Premier League visitors, who only made four changes from their last league outing, failed to test the committed Shrews after a limp display.
But the hosts were unable to find a way through to ensure David Moyes, a former Town player, avoided another embarrassment after suffering a shock as Everton boss in 2003 when Shrewsbury beat the Toffees 2-1.