Despite their recognized limitations, bibliometric assessments of scientific productivity have been widely adopted. We describe here an improved method to quantify the influence of a research article by making novel use of its co-citation network to field-normalize the number of citations it has received. Article citation rates are divided by an expected citation rate that is derived from performance of articles in the same field and benchmarked to a peer comparison group. The resulting Relative Citation Ratio is article level and field independent and provides an alternative to the invalid practice of using journal impact factors to identify influential papers. To illustrate one application of our method, we analyzed 88,835 articles published between 2003 and 2010 and found that the National Institutes of Health awardees who authored those papers occupy relatively stable positions of influence across all disciplines. We demonstrate that the values generated by this method strongly correlate with the opinions of subject matter experts in biomedical research and suggest that the same approach should be generally applicable to articles published in all areas of science. A beta version of iCite, our web tool for calculating Relative Citation Ratios of articles listed in PubMed, is available at https://icite.od.nih.gov.