A Chester team featuring one of their new additions, former Morecambe midfielder Alex Kenyon, had an early chance when Matty Williams glanced a corner off target.
Kurt Willoughby spurned another opportunity before Morecambe began to enjoy the better of proceedings.
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They came close to taking the lead on 15 minutes when Jon Obika rounded Chester keeper Harry Tyrer, only to see his effort cleared off the line by Harrison Burke.
Hunter chipped wide and sent a free-kick against the woodwork but opened the scoring midway through the half, collecting Dylan Connolly’s pass before finding the top corner from 15 yards.
Obika and Ryan Cooney saw attempts deflected behind before half-time, which saw Morecambe make three changes with another five coming on the hour.
Chester’s Joe Lynch sent a chipped effort onto the roof of Adam Smith’s goal early in the second half.
Weeks dragged a shot off target but made no mistake on 56 minutes though, beating a couple of players before firing past the Morecambe keeper.
The Shrimps responded with one of their half-time arrivals, Caleb Watts, hitting the woodwork.
Another second-half arrival, Cole Stockton, fired straight at Chester’s substitute keeper, Wyll Stanway, from distance.
Chester also sought a winner with James Horsfield firing over and Smith saving Anthony Dudley’s attempt.
The keeper also held Kole Hall’s late chip, while at the other end, Cooney was thwarted by a combination of keeper and defender as honours ended even.
Next up for the Shrimps is Tuesday’s return to the Mazuma Stadium, when they host Middlesbrough (7pm).
Chester FC: Tyrer (Stanway 46), Morgan, Heywood, Burke, Weeks, Murray (Lynch 46), Earl (Dudley 46), Willoughby (Hall 46), Stephenson (Marsh-Hughes 71), Williams, Kenyon (Horsfield 62). Subs not used: Clarke, Turner, Daly.
Morecambe: Smith, Cooney, Gibson (Melbourne 46), Fane (Taylor 46), O’Connor (Rawson 61), McLaughlin (Delaney 61), Connolly (Gnahoua 61), McLoughlin (Love 61), Obika (Stockton 61), Hunter (Watts 46), McDonald. Sub not used: Ripley.