Consumption of antimicrobial drugs, such as antibiotics, is linked with antimicrobial resistance. Surveillance of antimicrobial drug consumption is therefore an important element in dealing with antimicrobial resistance. Many countries lack sufficient surveillance systems. Usage of web mined data therefore has the potential to improve current surveillance methods. To this end, we study how well antimicrobial drug consumption can be predicted based on web search queries, compared to historical purchase data of antimicrobial drugs. We present two prediction models (linear Elastic Net, and non-linear Gaussian Processes), which we train and evaluate on almost 6 years of weekly antimicrobial drug consumption data from Denmark and web search data from Google Health Trends. We present a novel method of selecting web search queries by considering diseases and drugs linked to antimicrobials, as well as professional and layman descriptions of antimicrobial drugs, all of which we mine from the open web. We find that predictions based on web search data are marginally more erroneous but overall on a par with predictions based on purchases of antimicrobial drugs. This marginal difference corresponds to $<1$\% point mean absolute error in weekly usage. Best predictions are reported when combining both web search and purchase data. This study contributes a novel alternative solution to the real-life problem of predicting (and hence monitoring) antimicrobial drug consumption, which is particularly valuable in countries/states lacking centralised and timely surveillance systems.