Pozzi, 24, led the way for most of his heat but was forced to dig deep in the final stretch after colliding with the penultimate hurdle. A photo finish was required to confirm his second-place finish in the race with a time of 13.50s, two-hundredths of a second ahead of Jamaica's Andrew Riley.
Meanwhile 26-year-old Clarke, who came fourth in the event at London 2012, finished third in his heat with a time of 13.55s to seal his progression to the next stage.
Jamaica's Omar McLeod was the fastest qualifier overall from the session with a time of 13.27, while there were three places for Team USA in the semis as Ronnie Ash, Devon Allen and Jeff Porter all breezed through.
The semi-finals and final of the event take place on Tuesday evening.