Observations of the redshifted 21-cm hyperfine line of neutral hydrogen from early phases of the Universe such as Cosmic Dawn and the Epoch of Reionization promise to open a new window onto the early formation of stars and galaxies. We present the first upper limits on the power spectrum of redshifted 21-cm brightness temperature fluctuations in the redshift range z = 19.8-25.2 (54-68 MHz frequency range) using 14 h of data obtained with the LOFAR-Low Band Antenna (LBA) array. We also demonstrate the application of a multiple pointing calibration technique to calibrate the LOFAR-LBA dual-pointing observations centred on the North Celestial Pole and the radio galaxy 3C220.3, We observe an unexplained excess of similar to 30-50 per cent in Stokes / noise compared to Stokes V for the two observed fields, which decorrelates on greater than or similar to 12 s and might have a physical origin. We show that enforcing smoothness of gain errors along frequency direction during calibration reduces the additional variance in Stokes I compared Stokes V introduced by the calibration on sub-band level. After subtraction of smooth foregrounds, we achieve a 2 sigma upper limit on the 21-cm power spectrum of Delta(2)(21) <(14561 mK)(2) at k similar to 0.038 h cMpc(-1) and Delta(2)(21) <(14886 mK)(2) at k 0.038 h cMpc(-1) for the 3C220 and NCP fields respectively and both upper limits are consistent with each other. The upper limits for the two fields are still dominated by systematics on most k modes.