In a seven-minute period early in the second half the pair, who joined Dan O'Hearne's side this week, drew United level after the hosts went into half-time with a two-goal lead at Wombwell.
Dobby, an experienced midfielder who dropped down a league from Huddersfield Town to rejoin the club where her career began, headed home a corner shortly after the restart before Ramsden, an exciting 16-year-old winger making the step up to open age after impressing at the Regional Talent Centre, charged through the Barnsley defence and beat the keeper to equalise for Leeds.
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With minutes left to play, Whites veteran Bridie Hannon scored a third goal for Leeds to claim a win which shortens the gap to second-placed Newcastle to eight points.
On Wednesday, United travelled to Barnsley hoping to repeat the success they enjoyed on the reverse, when the Whites kept a clean sheet and put five past their South Yorkshire counterparts at the Global Stadium in September.
This time around, the Whites' clean sheet lasted just four minutes before Ruby Jex-Oldfield took the lead for the hosts, rounding off neat link-up play between Natalie Shaw and Natalie Watson with a first-time strike from 20 yards which flew well beyond the reach of Whites 'keeper Lauren Joyce into the back of the net.
Emily Pierrepont doubled the hosts' lead at the half-hour mark, stealing possession inside the final third and linking up with Shaw who provided another assist for the Barnsley number nine to turn home from close range.
Hannon and Ramsden both came close for the Whites before the teams went into half-time with the score at 2-0.
Leeds clawed one back straight after the interval as Dobby fired in a powerful header from a corner in the 47th minute to make it 2-1.
Seven minutes later, Ramsden pulled the Whites level with an impressive solo effort. Showing no fear, the young winger chased a through ball, holding off two Reds defenders as she raced into the box and tucked the ball into the far post.
Fast feet from substitute Aaliyah Nolan forced a frustrated Barnsley to concede a free-kick on the right wing with five minutes left to play. Hannon stepped up and confidently dinked the ball into the far post to give Leeds the lead.
United held on throughout five minutes of stoppage time before the referee's whistle confirmed the visitors' unlikely comeback.
After the game, match-winner Hannon explained how the result will give her side confidence going forward.
"Massively," Hannon said.
"It shows good character from the team, so I think we need to take the positives out of it."
With the Whites trailing by two goals to a side that have only claimed three league wins this season, half-time was a turning point in the clash, Hannon said.
"A game of two halves I'd say," Hannon told LUTV.
"We went in at half time 2-0 down. I don't know how we found ourselves in that position - we'd just not woken up
"We had a talking to at half time, as you can imagine and second half we were by far the better team, I don't think they had a shot on target.
"We said at half time, in order to win this game we are going to have to score an early goal.
"We just kept on plugging away, we had most of the possessions, we had a couple of chances - we didn't really test the keeper that much, to be honest - but I thought we dominated that second half.
"It was one of those Yorkshire derbies - they had more fight and desire in that first half, but I think the second half we showed who were and we played better football so I'm just so glad we got the three points in the end."