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The League Two hosts dominated their Championship opponents in the first half, taking the lead through Uche Ikpeazu's low turn and shot.
But a different Leeds came out after the break, equalising when Stuart Dallas nodded in Alex Mowatt's cross.
Mowatt then turned goalscorer when he flicked in from a corner to set up a tie at AFC Wimbledon or Sutton United.
The U's nearly equalised late on when Piero Mingoia's shot was spilled by Marco Silvestri but Ben Williamson could not turn the ball home under pressure.
In front of a sell-out crowd, two sides going for promotion in their respective leagues provided an entertaining contest, despite the 38 places separating the teams.
Ikpeazu had an earlier chance to score in the game but had an effort saved, before Luke Berry's free-kick deflected back to him, he turned two defenders and struck low in off the post.
Leeds twice forced Will Norris into fine saves in the first period from Mowatt and Liam Cooper.
But Garry Monk's side, in the Championship play-off places and having made eight changes for the match, showed fight in the second half and equalised when Dallas knocked Mowatt's cross back across the face of goal.
Mowatt then completed the turnaround when he flicked on Pontus Jansson's header from a corner into the roof of the net.
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Cambridge, managed by former Leeds midfielder Shaun Derry, had chances to force a replay, but substitute Williamson could not turn home and Ikpeazu headed a corner wide.
Cambridge United manager Shaun Derry: "We want more of this. We want to be on a different stage.
"We are in League Two and it's a tough, tough level and we can make a real claim for the top seven now."
Leeds boss Garry Monk: "Cambridge made it very tough for us as we knew they would. They were excellent in that first half, they put pressure on us and were a handful.
"We were able to move the game up the pitch in the second half and got the goals at good times.
"I trust all my players - there's no point in having a squad if you are not going to use them and it will stand them in good stead.
"I love the FA Cup, I've grown up with it and it means a lot to me. All I know is that we use our squad, we respected it and you could see that from how committed we were."
Steve Claridge, BBC Radio 5 live co-commentator and former Cambridge United striker:
"Lots of things that have gone on at Leeds have left a slightly bitter taste. But what they done is given the manager a chance and backed him financially. You get the feeling Garry Monk, with what he has had to put up with, just lets it wash over him. He comes across as a focused person.
"Leeds have the capabilities of being a top six or seven team in the country. They will get 35,000 every home game if they win promotion to the Premier League. They have the ammunition and quality to give it a real go."
Match ends, Cambridge United 1, Leeds United 2.
Second Half ends, Cambridge United 1, Leeds United 2.
Foul by Alex Mowatt (Leeds United).
James Dunne (Cambridge United) wins a free kick on the left wing.
Attempt missed. Kemar Roofe (Leeds United) right footed shot from more than 35 yards is high and wide to the right. Assisted by Pontus Jansson.
Foul by Stuart Dallas (Leeds United).
Blair Adams (Cambridge United) wins a free kick in the defensive half.
Corner,  Leeds United. Conceded by Luke Berry.
Attempt missed. Uche Ikpeazu (Cambridge United) left footed shot from outside the box is high and wide to the left. Assisted by Greg Taylor.
Attempt saved. Piero Mingoia (Cambridge United) right footed shot from outside the box is saved in the centre of the goal. Assisted by Uche Ikpeazu.
Substitution, Cambridge United. Ben Williamson replaces Max Clark.
Lewie Coyle (Leeds United) wins a free kick on the right wing.
Foul by Harrison Dunk (Cambridge United).
Lewie Coyle (Leeds United) wins a free kick in the defensive half.
Foul by Harrison Dunk (Cambridge United).
Alex Mowatt (Leeds United) wins a free kick on the left wing.
Foul by Greg Taylor (Cambridge United).
Attempt missed. Uche Ikpeazu (Cambridge United) header from very close range misses to the left. Assisted by Max Clark with a cross following a corner.
Corner,  Cambridge United. Conceded by Tyler Denton.
Attempt blocked. Max Clark (Cambridge United) right footed shot from outside the box is blocked.
Foul by Kalvin Phillips (Leeds United).
Luke Berry (Cambridge United) wins a free kick on the right wing.
Attempt blocked. Kalvin Phillips (Leeds United) right footed shot from outside the box is blocked. Assisted by Kemar Roofe.
Tyler Denton (Leeds United) wins a free kick in the defensive half.
Foul by Piero Mingoia (Cambridge United).
Attempt missed. Marcus Antonsson (Leeds United) header from the centre of the box misses to the right. Assisted by Kemar Roofe with a cross.
Kemar Roofe (Leeds United) wins a free kick on the left wing.
Foul by James Dunne (Cambridge United).
Offside, Cambridge United. Max Clark tries a through ball, but Mark Roberts is caught offside.
Foul by Kyle Bartley (Leeds United).
Uche Ikpeazu (Cambridge United) wins a free kick in the attacking half.
Attempt missed. Alex Mowatt (Leeds United) left footed shot from the left side of the box misses to the right. Assisted by Kemar Roofe.
Delay over. They are ready to continue.
Substitution, Leeds United. Kyle Bartley replaces Liam Cooper because of an injury.
Delay in match Liam Cooper (Leeds United) because of an injury.
Liam Cooper (Leeds United) wins a free kick in the defensive half.
Foul by Uche Ikpeazu (Cambridge United).
Substitution, Cambridge United. Harrison Dunk replaces Conor Newton.
Goal!  Cambridge United 1, Leeds United 2. Alex Mowatt (Leeds United) header from very close range to the top left corner. Assisted by Pontus Jansson following a corner.
Attempt blocked. Pontus Jansson (Leeds United) header from very close range is blocked. Assisted by Stuart Dallas with a cross.
Leeds United came from behind to defeat stubborn Cambridge United in a hotly contested FA Cup third-round tie.