Orient looked like they would cruise to an easy victory after McGleish and Alex Revell both scored within the first 25 minutes, but two goals in as many second-half minutes from Danny Green and Solomon Taiwo drew the visitors level with 20 minutes remaining.
But McGleish scored again with 14 minutes remaining to ensure Orient’s passage into the second round.
Elsewhere, League One leaders Brighton and Hove Albion survived a scare, needing penalties to sneak past Conference South side Woking.
The replay finished 2-2 after Brighton had surrendered an early lead, and the favourites were forced to come from a goal behind in extra time and play for 15 minutes with 10 men before eventually winning on penalties.
League One side Huddersfield were lucky to survive against Conference side Cambridge United, scoring two goals in injury time to steal a 2-1 victory.
Lee Peltier and Gary Roberts were the heroes for Huddersfield, with their late strikes cancelling out Rory McAuley’s 53rd-minute goal.
Former Premier League side Charlton Athletic are also through to the second round after a Kyel Reid goal gave them a 1-0 win at home against Barnet, while Port Vale thumped Dartford 4-0 thanks to a hat-trick from Marc Richards.
Macclesfield Town needed penalties to defeat League Two rivals Southend United after their replaye finished locked at 2-2 after extra time.
Peterborough had a much easier time of things against Stockport United, winning 4-1, while a Reuben Reid double gave Walsall a 2-0 win over Fleetwood Town.
Stevenage dumped Milton Keynes Dons out of the Cup, winning 6-4 on penalties after their replay was locked at 1-1.
Meanwhile, Hartlepool and Aldershot Town both moved through to the second round with 1-0 wins over Vauxhall Motors and Brentford respectively.
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