Despite a tumultuous recent period at Tynecastle, Hearts moved into the top half of the table in sixth place, above United, as Stevenson scored twice in the first half, the second a penalty.
McGlynn, whose side next play at Kilmarnock on Boxing Day, said: "We're delighted that over the start of these four games coming up we've won the first. We go down to Rugby Park on Wednesday, which will be difficult.
"It's nice to get three points on the board and go down there with a feel-good factor and some confidence."
Hearts' Edinburgh rivals Hibernian have been praised for turning from relegation candidates last term to top-six hopefuls this season, yet McGlynn's men are only three points behind.
McGlynn hopes Hearts can overtake Hibs come the Edinburgh derby early in the New Year.
"We'd like to be in that position come January 3," he said.
- Sports & Recreation