Dagnall scored early in both halves with Marcus Maddison equalising in between.
Bradley Pritchard's clever flick on the hour sealed the win for Orient, who played over 20 minutes with 10 men when Romain Vincelot was sent off, before Posh debutant Luke James scored a late consolation.
Substitute Tubbs scored the decisive kick to maintain a perfect record for the hosts in the shoot-out after Southend's Lee Barnard struck a post and Dons goalkeeper James Shea saved Ben Coker's spot-kick.
League Two Morecambe pulled off a shock at Highbury, beating last season's Northern Area finalists Fleetwood 3-1.
Jordan Hugill's second-half goal was enough to see Preston avoid a cup upset with a 1-0 win over Shrewsbury.
Last season's finalists Chesterfield bowed out in the first round this time in a 2-0 defeat at Scunthorpe.
John Marquis and Koby Arthur scored the goals to take Cheltenham through to the second round as the Robins defeated Oxford 2-0 at Whaddon Road.
Devante Cole and Kane Hemmings scored second-half goals as Barnsley beat York 2-0.
Luke Norris' first-half penalty sent 10-man Gillingham into round two as they beat Stevenage 1-0 at the Lamex Stadium.
Notts County proved too hot to handle for local rivals Mansfield as the Magpies secured a 2-0 win at Meadow Lane.
Managerless Carlisle celebrated their first win of the season with a 3-1 victory over Accrington, who are still waiting for their first victory of the campaign.
Elsewhere, there were wins for Rochdale, Oldham, Crawley, Coventry and Portsmouth.
- Sports & Recreation
- AFC Wimbledon
- Romain Vincelot
- Matt Tubbs
- Bradley Pritchard
- Lee Barnard