Summary The two most commonly used airway management techniques during general anaesthesia are supraglottic airway devices and tracheal tubes. In older patients undergoing elective non‐cardiothoracic surgery under general anaesthesia with positive pressure ventilation, we hypothesised that a composite measure of in‐hospital postoperative pulmonary complications would be less frequent when a supraglottic airway device was used compared with a tracheal tube. We studied patients aged ≥ 70 years in 17 clinical centres. Patients were allocated randomly to airway management with a supraglottic airway device or a tracheal tube. Between August 2016 and April 2020, 2900 patients were studied, of whom 2751 were included in the primary analysis (1387 with supraglottic airway device and 1364 with a tracheal tube). Pre‐operatively, 2431 (88.4%) patients were estimated to have a postoperative pulmonary complication risk index of 1–2. Postoperative pulmonary complications, mostly coughing, occurred in 270 of 1387 patients (19.5%) allocated to a supraglottic airway device and 342 of 1364 patients (25.1%) assigned to a tracheal tube (absolute difference −5.6% (95%CI −8.7 to −2.5), risk ratio 0.78 (95%CI 0.67–0.89); p < 0.001). Among otherwise healthy older patients undergoing elective surgery under general anaesthesia with intra‐operative positive pressure ventilation of their lungs, there were fewer postoperative pulmonary complications when the airway was managed with a supraglottic airway device compared with a tracheal tube.