Having drawn 1-1 in their first meeting on November 8, Telford - who are bottom of their league - came through a replay against the Conference South team thanks to first-half goals from Andrew Parry and Tony Gray.
Louie Soares halved the deficit seven minutes after the break and Basingstoke were given further hope when Telford's Michael Grogan was sent off in the 57th minute.
However, the visitors failed to find a leveller and were reduced to 10 men themselves when Jamie Brown saw red in second-half injury time.
Telford can look forward to a visit to Ashton Gate, while fellow Conference Premier strugglers Southport are through to round two for the first time in 12 years after surprising League Two side Dagenham and Redbridge 2-0 at Merseyrail Community Stadium.
John Marsden's 50th-minute opener and a penalty from Danny Hattersley did the damage for the hosts, after Luke Howell was shown a straight red card for the visitors a minute before the break.
Sheffield United set up a meeting with Plymouth Argyle by beating Crewe Alexandra 2-0 at Bramall Lane, while AFC Wimbledon and Rochdale were 3-1 and 2-1 victors at home to York City and Northampton Town respectively.
League Two sides Accrington Stanley and Shrewsbury Town defeated League One opposition in the shape of Notts County and Walsall to secure their passage.
In the two rearranged first round fixtures, Doncaster Rovers safely saw off a potential banana skin by beating Weston-super-Mare of the Conference South 4-1, while Concord Rangers - also from the Conference South - earned a replay by drawing 1-1 at Mansfield Town, with Cambridge United awaiting the victors.
