Rapidly evolving pathogens pose a challenge to vaccine design, as their mutations render previous vaccine responses obsolete. For influenza, conserved epitopes on the viral coat proteins have been identified, but mysteriously they are missed by the antibodies elicited by most vaccine recipients [1, 2]. In simulated immunizations using 263 million antibody-hemagglutinin (HA) structural docking solutions, non-conserved epitopes were immunodominant when HAs were immunized at standard concentrations. However, by vaccinating with a pool of 30 diverse and dilute HA variants, B-cells that recognize broadly-conserved epitopes across HA receive up to 30-fold higher antigen dose, with concentration being linearly correlated to conservation on a per epitope basis. If individual variants are at concentrations below the minimum threshold of immune activation, then cross-reactive B-cells will be preferentially elicited. In pig immunizations, the approach induced a broad-spectrum antibody response against a panel of 36 strains from 1918-2014, including all pandemic strains from the past century and multiple strains not in the vaccine. In further swine studies with a vaccine containing HAs from 1918-2008, we observe broad-spectrum neutralizing responses against 6 future heterologous strains, including pandemic strains, spanning H1N1 2009-2017 and H3N2 2009-2014. Our results support a greater understanding of why non-conserved epitopes are immunodominant, as well as indicate a general solution to overcome this in broad-spectrum vaccine design.