Russell Slade's men extended their unbeaten away run in the league to six games courtesy of goals either side of half-time from Moses Odubajo and Scott Cuthbert.
The victory lifts Orient above Wolves into second in the table, just one point adrift of leaders Brentford.
Struggling Shrewsbury, meanwhile, have now lost five League One home games in a row and remain only two points clear of the relegation zone.
Graham Turner's side spurned the first clear-cut chance of the match when Aaron Wildig sliced wide after being put through on goal by Tom Eaves.
And Wildig's wastefulness was punished just before the half-hour mark as Odubajo took full advantage of a mix up between keeper Chris Weale and his defence to slot home his sixth League One goal of the season.
Eaves himself then missed a golden opportunity to equalise shortly after the break when he fired just wide as the hosts pushed for a leveller.
However, Orient showed exactly why they are pushing for promotion by sealing a hard-fought three points when Dean Cox's corner was diverted past Weale by Cuthbert's knee four minutes from time.
