The home side stunned the Yorkshire outfit with three tries in the opening few minutes, No 8 Anthony Maka, scrum-half Jake Garside and centre Matt Hema all touching down – but crucially none of the conversions were made because of a swirling wind.
Tykes got off the mark before the break with full-back Andrew Lawson racing in and centre Charlie Venables adding the conversion. Suddenly, the game looked very much there for the taking.
With the wind mostly on their backs after the break Leeds dug in, and some strong defensive work ensured Cambridge were held out.
Prop Xavier Valentine, inset, crashed over around the hour mark and Venables converted to close to within one point at 15-14, and when Cambridge were caught offside, Venables’ penalty proved the difference between the two sides.
Wharfedale dropped to third bottom of National Two North after losing 24-13 at Tynedale.
Right winger Hamilton scored their only try just after half-time with centre Davidson kicking the conversion and two penalties, but it wasn’t enough to catch Tynedale.
Unbeaten Otley remain top of the North Premier table despite their game at Alnwick falling victim to the weather.
York missed the chance to climb the table after losing out 49-24 at Blackburn, while Sandal were also left without a match after their scheduled game at Kirkby Lonsdale in Cumbria was postponed.
Leaders Driffield secured a 38-14 victory at Bradford and Bingley to consolidate their lead. Old Brodleians are now second after losing out by the narrowest of margins at third-placed Heath, 19-18.
Meanwhile, Pocklington were 20-7 winners over Cleckheaton to boost their title hopes.