He does what Tilly should have done. Realizes he's in over his head, and immediately attempts to give command to a more experienced officer.
Harriman was also stymied by the fact that the Enterprise-B was put into an emergency rescue operation without basic necessities (tractor beam, photon torpedoes, a medical staff). The Discovery-A presumably has the full capability of 32nd century technology.
Tilly should have been commanding night shifts and probably holodeck training and other minor situations, so that her being put in command during Saru's away mission would be old hat for the bridge crew. And give her a second. If something comes up, as it did, have Nilsson there (or better yet, Burnham) to take over. No harm, no foul, better luck next time.