Having made a first League One appearance off the bench, followed by an EFL Trophy start, the on-loan Fulham striker, plagued by injury this season, played off Kieffer Moore in the FA Cup against County and he struck in the 48th minute against Harry Kewell's visitors.
Left-back Zeki Fryers, one of 10 changes for the midweek EFL Trophy tie against Everton Under-21s, had done enough to keep his place and he netted the second soon after Woodrow's opener.
Brad Potts and Moore ensured there would be no comeback by netting in the 77th and 81st minutes in front of a 5,878 all-pay crowd.
Barnsley had selected a much stronger line-up for the game's premier knockout competition while League Two strugglers County were unchanged from the side which earned a goalless home draw against Oldham in midweek.
Early on, Moore volleyed wide for Barnsley but Nathan Thomas sidefooted off target from 14 yards after being set up by Enzio Boldewijn following a swift counter-attack.
Ross Fitzsimons claimed a downward header from Moore, who followed up by back-heeling the ball into the are for Woodrow to race on and force a good save from the goalkeeper.
An overhead shot from County's Jamie Turley flew wide and Moore skied a shot from 10 yards as the half ended goalless.
Barnsley soon went ahead after the break, Cameron McGeehan's through ball seeming to be helped on by Elliott Ward, allowing Woodrow to race in at the far post and get beyond Jamie Turley to volley home.
It was 2-0 five minutes later in the 53rd when the Reds moved the ball from left to right and a low cross from Brad Potts was sidefooted home from inside the area by Fryers.
Thiam's fierce drive forced a save from Fitzsimons as the Reds looked for a third before Woodrow and Dimitri Cavare gave way to Mike Bahre and Jacob Brown.
Kion Etete and Dan Jones replaced Kristian Dennis and Thomas for the visitors.
A stretching Potts went close to converting Moore's low cross before the pair combined again and this time Potts tapped home from inside the six-yard area.
It could easily have been 4-0 as Bahre exchanged passes with Potts on the counter before seeing his left-footer pushed aside for a corner.
The Reds would not be denied and when a corner was cleared back to Thiam he chipped the ball back across for an unmarked Moore to dive forward and head home inside the six-yard area.
Moore's rasping drive went close to making it 5-0 and his close-range effort was also turned onto the woodwork as Barnsley finished in command.