The match had been heading for a stalemate but Grant Holt headed home the winner in the 78th minute to maintain his side's 100% record.
The Dags, meanwhile, could only draw 1-1 at home to 10-man Port Vale, allowing Bradford and Bury to move level
on nine points with the Shrews in second and third.
Paul Benson gave the home side a fourth-minute lead which was quickly cancelled out by Marc Richards' effort three minutes later.
Dagenham pushed for a winner - helped by Sam Stockley's 79th-minute dismissal - but had to settle for a point.
Bradford won 2-0 at home to Rochdale with Michael Boulding's 20th-minute header and Peter Thorne's strike after 34 minutes doing the damage.
Lancastrians Bury, meanwhile, beat Morecambe 2-1 at Gigg Lane.
Rene Howe had given Morecambe a 14th-minute goal but the Shakers hit back through Paul Scott and David Artell's own goal.
Paul Mullin and James Ryan scored to help Accrington to a 2-0 win at home to Macclesfield.
Craig Woodman's 15th-minute goal was enough to give Wycombe a 1-0 win over Lincoln City to keep them two points behind the leaders.
Barnet midfielder Albert Adomah missed a second-half penalty as they lost 1-0 at home to Brentford. Glenn Poole's 10th-minute spot-kick had given the Bees the lead.
Ryan Harley's 68th-minute goal earned Exeter a 1-0 win at Bournemouth while Jamie Ward scored in Chesterfield's 1-0 win at Grimsby.
Crisis-hit Luton drew 1-1 at home to Notts County and Rotherham, like the Hatters hit with a points deduction before the season started, beat Chester 3-1 at home.
Garry Richards scored an injury-time winner as Gillingham triumphed at Darlington's expense in the north east.
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