City boss Billy McEwan failed in his attempt to bring in an experienced keeper to deputise for suspended first-choice keeper Tom Evans. As a result youngster Aaron Reid was handed his full debut. Otherwise the Minstermen's line-up was unchanged to the side that was beaten 3-0 at the weekend.
Visitors Morecambe also suffered a 3-0 defeat at the weekend and boss Sammy McIlroy brought back first-choice keeper Steven Drench, who had recovered from injury. The only other chance saw Ged Brannan return to central defence.
City started the better and created a number of early chances. Inside the first minute Clayton Donaldson's cross was headed out for a corner by Dave Perkins. Then on five minutes Craig Farrell had the chance to open his City account from Neal Bishop's flick but shot straight at Drench.
Martyn Woolford, making his first full start in front of home fans, showed great skill to skip past Adam Yates inside the box but Drench brought off a fine save at his near post from the angled shot. As City continued to press, Emmanuel Panther was denied his first City goal when Drench parried his 20-yard drive.
The visitors' first chance came courtesy of a Craig Stanley cross which only just eluded Wayne Curtis at the far post. Moments later Morecambe went ahead when a right-wing cross was blocked by James Dudgeon but the ball fell invitingly to Twiss who shot home from eight yards.
City were clearly rocked by going behind and defensive frailties began to show before Garry Hunter was inches over from 25 yards. Morecambe doubled their lead on 40 minutes with a training-ground free-kick paying dividends. Stanley laid the ball back into the path of Gary Thompson who side-footed home with pinpoint accuracy from 20 yards.
In first-half stoppage time top-scorer Donaldson attempted a spectacular overhead effort which flew over the bar. At the other end a terrible back-header by Darren Craddock gave Curtis the chance to wrap up the points but his lobbed effort went wide.
City were back in the game with a goal three minutes after the restart. Donaldon's run and cross from the right picked out Woolford who volleyed home at the post.
Seconds later the recent signing almost scored a second from Farrell's excellent pass. Then Farrell himself was denied his first City goal when Drench gathered his close-range effort from Mark Convery's nod down.
Visitors Morecambe created chances of their own with Danny Carlton guilty of missing when well placed inside the box.
City defender Craddock was fouled by Perkins in the box on 75 minutes and Donaldson calmly despatched the resulting penalty for the equaliser.
City's joy turned to misery within 60 seconds when Morecambe regained the lead as Twiss scored his second of the night.
In a ding-dong finish Convery saw his diving header strike Perkins and bounce to safety.
Then in stoppage time defender David McGurk was denied an dramatic equaliser when he shot straight at Drench in a crowded box from a long throw-in.