United were the livelier of the two sides during the opening exchanges and went on to dominate play during an entertaining first half, leading 3-1 at the interval.
The deadlock was broken in the sixth minute when the unfortunate Bedford captain Kyal McNulty diverted a dangerous low cross into his own net when attempting to clear.
On 15 minutes Greenford extended their lead as Eagles keeper Maxime Fondop-Siwe failed to gather cleanly and Aaron McLeish drove a clinical low finish into the corner.
Six minutes later, against the run of play, Bedford got themselves back into the game when Afolabi Obafemi placed an accurate shot past United keeper Josh Kensit.
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United’s two-goal cushion was deservedly restored on the half hour mark, however, when Greenford captain Luke Kanjor unleashed an unstoppable left-footed strike into the top corner of the net.
The Eagles raised their game after the interval in their efforts to reduce the deficit.
But led by Nick Turner and Liam Hind at the back, United protected their first-half lead with determined, resolute, no-nonsense defending.
In the end Kensit was not seriously tested, while on the stroke of full-time, it was United who came closest to grabbing a goal when substitute Ike Nzuruba missed a great opportunity to add a fourth.