Pulis left the club on Monday night having been in charge for just 45 days and 10 games – and less than 24 hours later the Owls recorded a victory under caretaker Neil Thompson.
The win was relegation-battling Wednesday's second in three games.
Middlesbrough manager Neil Warnock made one change from the team that beat Birmingham 10 days ago, with goalscorer Watmore one of those brought in.
Wednesday took the lead five minutes later when Liam Palmer found Paterson, who fired the ball into the bottom right corner.
The hosts doubled their lead in the 40th minute as Shaw scored his first senior Wednesday goal by tapping in from a Bannan cross.
Warnock's visitors were quick out of the blocks after the interval and Watmore pulled one back in the 49th minute. The Boro winger latched on to a Sam Morsy pass and found the back of the net.
Wednesday almost restored their two-goal advantage in the 69th minute when Paterson flicked a Windass pass on target but Boro goalkeeper Marcus Bettinelli made the save.
Boro, chasing a fourth successive victory, pressed for an equaliser and substitute Marcus Tavernier had a good chance in the 71st minute but could only fire wide of the Wednesday goal.
The chances kept coming for Boro, with Johnson rattling a post a minute later.
Warnock threw on Marcus Browne and he had Boro's last chance of an equaliser but he hit low and wide of Westwood's goal as Wednesday held on for victory.