Barrow, Forest Green, Blyth Spartans, Torquay United and Kidderminster Harriers are the other sides from outside the Football League to go into Sunday's third round draw.
With Wycombe top of League Two few would have given the Unibond Premier League side Eastwood much hope of an upset, even with home advantage.
But Lindon Meikle put them ahead with a superb curled shot from 20 yards, and Peter Knox sealed the win with a close range effort.
Torquay were comfortable 2-0 victors over Oxford United at Plainmoor, while Kidderminster saw off Curzon Ashton by the same score.
Forest Green defeated the League Two side Rochdale, also 2-0.
Marc Bell pulled off a string of fine saves to secure Blyth a replay against Bournemouth, meaning their long trip to the south coast was not a wasted journey.
Bell, who works as a roofer during the week, denied Lee Bradbury three times to be the hero for the Blue Square North side.
A Matt Fryatt hat-trick guided Leicester City into the third round after they had twice fallen behind to the League Two side Dagenham and Redbridge.
A hat-trick of headers - two from Gary Alexander and one from Ashley Grimes - ensured there was no fairytale return to the New Den for Gary Waddock as Millwall beat Aldershot 3-0.
Leyton Orient, Hartlepool United, Scunthorpe United, Southend United are all through.
Carlisle United beat Grays Athletic 2-0 in a rescheduled first round replay.
Barrow, from the Blue Square North, were 2-1 victors over Brentford at Holker Street on Friday night.
David Brown put them ahead from the penalty spot after a foul from the visiting keeper Ben Hamer, who was sent off for the offence.
Although Charlie McDonald levelled for the London side, Matt Henney got a late winner to send the Cumbrian non-leaguers through.
Macclesfield Town beat Port Vale 3-1 at Vale Park.
