The first evidence of spin alignment of vector mesons (K∗0 and φ) in heavy-ion collisions at the Large Hadron Collider (LHC) is reported. The spin density matrix element ρ00 is measured at midrapidity (|y|<0.5) in Pb-Pb collisions at a center-of-mass energy (sNN) of 2.76 TeV with the ALICE detector. ρ00 values are found to be less than 1/3 (1/3 implies no spin alignment) at low transverse momentum (pT<2 GeV/c) for K∗0 and φ at a level of 3σ and 2σ, respectively. No significant spin alignment is observed for the KS0 meson (spin=0) in Pb-Pb collisions and for the vector mesons in pp collisions. The measured spin alignment is unexpectedly large but qualitatively consistent with the expectation from models which attribute it to a polarization of quarks in the presence of angular momentum in heavy-ion collisions and a subsequent hadronization by the process of recombination.