Newport were a goal down at the interval but two minutes into the second half, Reid sent a powerful head past home keeper Harrison for his seventh goal of the season. Gary Warren hit a second eight minutes later to give Dean Holdsworth side the victory.
Lee Hughes scored for Notts County in a 4-2 League Cup defeat at Wolves and Robbie Keane scored for Spurs in a 4-1 home defeat to Arsenal.
Brentford's penalties win against Everton after a 1-1 draw and Burnley's 1-0 win over Premier League Bolton were the only shocks.
Kevin Kyle scored for Hearts in a Scottish League Cup defeat at Falkirk.
Locally, Stratford Town beat Oadby Town 2-1 in a Midland Alliance fixture.
In the FA Vase replays, Coventry Copsewood are through to the First Round Proper after a 4-1 victory over Dudley Town at Allard Way but Southam United are out after a 3-0 defeat at Thurnby Nirvana.