Bolton striker Eaves signed on loan for a second spell at New Meadow on Thursday after scoring six goals in 10 games for the club at the end of last season.
He was joined on the scoresheet by Adam Reach and Joe Jacobson, and they all helped to overturn Francois Zoko's 24th-minute opener for the hosts.
Zoko - Boro's leading scorer - grabbed his seventh of the season after 25 minutes, tapping in the rebound after Filipe Morais had rattled the post from 25 yards.
But the equaliser came 10 minutes later, Reach lifting over on-rushing keeper Chris Day after Luke Summerfield had ghosted through the Boro midfield.
The Shrews were soon ahead when Eaves nodded home Paul Parry's cross seven minutes before the break.
Jacobson increased the lead early in the second half when he headed home Darren Jones' cross and the defender went on to clear Morais' header off his own line minutes later.