There was not much to choose between the teams during the first half. However, the visiting Bluebirds broke the deadlock in the 32nd minute.
The goal had a huge element of fortune about it, John Rooney shooting from the just outside the Harrogate box and seeing his strike take a huge deflection before looping past a wrong-footed James Belshaw.
The second period belonged to the hosts and a quick-fire triple put them in control.
They restored parity eight minutes after the resumption, Joe Leesley whipping in a ball from the left that narrowly evaded Jack Muldoon, but was diverted into his own net by Kyle Jameson.
Moments later Muldoon twisted and turned inside the box before rifling an effort across and in past Andrew Firth.
Barrow looked shell-shocked and when captain Falkingham seized upon a loose clearance by Firth and scored from the edge of the area in the 58th minute the hosts were in full control.
On 69 minutes Mark Beck stabbed home Leesley’s teasing left-wing free-kick although the visitors rallied in the closing stages and pulled one back courtesy of Tyler Smith’s glancing header.
FC Halifax Town battled to a point at high-flying Solihull as they drew 0-0 at Damson Park.
The two sides went into the game in contrasting form, with Halifax winless in the league in 10 games, and Solihull winning six out of their previous seven.
Halifax survived a first-half barrage, including Solihull missing a penalty, to make more of a game of it after the interval.
Sam Johnson was forced into a superb double save in the early minutes from Jamey Osborne and Danny Wright, which set the tone for the opening half, as a combination of wasteful finishing and excellent goalkeeping kept Solihull at bay.
Danny Wright squandered a golden opportunity around the half-hour mark when he blazed a penalty over the bar after Jermaine Hylton had raced clear and dribbled round Johnson before being fouled by Matty Brown.
Hylton then thought he had scored soon afterwards but his header was deemed not to have crossed the line.
In National League North, Bradford Park Avenue continued their push for promotion with a 2-2 draw at home to Darlington, but there were defeats for both York City and Guiseley.
The Minstermen lost 3-0 at Spennymoor Town, while the Lions went down 3-1 at home to Blyth Spartans.