In truth the outcome was rarely in doubt from the moment debutant Asa Hall headed the home side in front after just four minutes - with Shrewsbury's advantage doubled just after the half-hour when striker Guy Madjo netted on his home debut.
There was scarcely a response from a well-below par Shrimps, who did not manage their first attempt on goal until the dying seconds of the first half.
With Town in control of the game, without playing particularly well themselves, the second half was decidedly low-key as the home side ran out easy winners.
Shrews boss Gary Peters had been active in the transfer during the past ten days with a home debut handed to striker Madjo - signed from Conference outfit Crawley Town - and a full debut given to midfielder Hall, who moved to Shrewsbury from Birmingham City this week in a loan deal for the rest of the season.
The decision to sign Hall paid immediate dividends as he was on hand to meet a Marc Pugh cross and head home from eight yards to give Town a fourth minute lead.
Morecambe, their play-off hopes dented by a 3-0 home defeat against Darlington in midweek, were unable to settle and lacked a goal threat and it was no surprise when Shrewsbury went further ahead on 31 minutes.
When the visitors defence were unable to clear another Pugh cross the ball fell to Madjo and from 12 yards he scored with a smart, low angled drive.
After the break the home side continued to press with efforts from Hall and Pugh requiring saves from visiting keeper Shawn Jalal.
At the opposite end a swerving cross from full-back Adam Yates struck the angle of crossbar and post in a rare Morecambe raid, but soon after coming on, Shrews substitute Andy Cooke almost made it three, but headed just over with seven minutes left.