Stead’s 84th-minute winner sealed a late victory for the Bees, having come from behind after Josh Gordon gave the struggling visitors a surprise lead in the 3rd minute.
Barnet restored parity on the stroke of half-time when Mark Shelton got on the end of Idris Kanu’s cross before Kabongo Tshimanga knocked a deflected effort past Barrow goalkeeper Wyll Stanway.
Brennan’s side then took the lead seven minutes into the second half through Mark Shelton before Rekeem Harper fired past Cieran Slicker to level the scores once more.
Talisman Stead then notched the winner as his deft header sailed past Stanway to secure a vital three points, pushing the Bees closer to an unlikely play-off charge.
“It’s that time of the season where it’s all about the result. We showed good character and good grit, and we’ve got that winning mentality back at the football club,” a gleaming Brennan told the club website post-match.
“I’m in my fourth full year here, and every year we’re pushing for promotion. We’re in that chase again. It’s a different chase to what we were in last year, but I thought the determination of the players lent itself to that winning mentality that we showed on the day. They (Barrow) made it difficult.
“We weren’t clinical enough at times, but the main thing was the victory, so I’m very pleased.”
Barnet were undone by an early setback when Gordon opened the scoring in the third minute with Cieran Slicker beaten at his near post.
“I thought Slicker was beaten a little too easy. I’m not critical of Slicks; he’s been amazing for us. He made a save at Fleetwood on Monday that was insane. I’m not looking to be overly critical of Slicks, but how we got in that position wasn’t great from our point of view,” said Brennan.
“It’s a good finish (from Gordon). You can give him a bit of credit, but we kept building the pressure.
But the Bees’ relentless nature – largely instilled by Brennan himself – allowed the play-off hopefuls a way back into the game just before half-time.
On the reaction to going behind so early, Brennan said: “We scored at a fantastic time. It was right on half-time with Shelton arriving as he does, and it’s gone in off Kabi’s backside, which he’ll take all day long.
“We started the second half a little bit sloppy for the first sixty seconds again. I thought they got in our areas a little bit too easily. We then built a little bit of momentum, and Stead has played a great pass for the second goal.
Seemingly irked by the way Barnet allowed Barrow back into the game when Harper brilliantly levelled, Brennan gave a nod to a fellow north London club and the impact that pressure can have this late into a season.
“I watched Arsenal earlier (2-1 home defeat to Bournemouth). Everyone is desperate for Arsenal to win the title. It shows you how difficult football is and how difficult it is to win football matches,” said Brennan.
“As the line gets closer, it shows you how nervous teams can get. It can happen to lads in division four as well at our level.”
With just five points separating Barnet from seventh-placed Chesterfield, Brennan leads his team to Notts Country this weekend (Saturday 3pm) before League Two fixtures against Gillingham and Harrogate Town as the club’s push for a play-off spot intensifies.