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The National League outfit opened the scoring after 22 minutes when Maz Kouhyar pulled the ball back to Lewis Walker and he fired home at the second attempt after his initial shot had hit the upright.
The Harriers striker continued to cause problems, putting an effort over the crossbar and another just wide, but in the 36th minute he turned provider, charging upfield before teeing up Kouhyar and the on-loan Notts County midfielder slotted home.
Doncaster’s only effort of the half came in added time when Alfie May’s chip had to be clawed away by home goalkeeper Jack Wint.
Rovers boss Grant McCann made changes at the break and substitute Leon Ayinde began to stretch the Harriers defence.
The visitors had calls for a penalty waved away when May went sprawling on the deck as he tried to connect with a Jordan Thomas pass inside the area, but two minutes later they fell further behind.
Kouhyar was again involved, having a shot that Rovers keeper Josh Clarke could only parry and David Davis turned in the rebound.
Late pressure saw Hakeeb Adelakun and Isaac Hutchinson test Wint but the keeper was up to the task.