The young winger, released by West Ham in the summer, was not expected to play a big part in the Glovers' early season games with Slade saying he would use him as a "late impact player".
But Welsh was given the full 90 minutes and scored what proved to be a brilliant winner just before half-time.
Promoted Carlisle had taken four points from their opening two games and had not conceded a goal. When they finally let one in, 197 minutes into their season and 17 on the day, it was a giveaway.
Nathan Jones floated a free-kick goalwards from the centre circle and when the Cumbrians defence dithered, the impressive Chris Cohen reacted quickly to head in.
Six minutes later it was 1-1 and Yeovil fans needed no introduction to the goalscorer. After four years with the Glovers, Kevin Gall was released and headed north to sign a two-year contract with Carlisle.
Always pacy, Yeovil let him go because of a lack of finishing, but he slipped quickly on to a perfect pass from the talented Michael Bridges to knock the ball past Steve Mildenhall.
What proved to be the winner came in the 42nd minute. Welsh and Jones tried a close passing movement on the edge of the box, but appeared to have lost the initiative. However, a rebound from a defender left the ball at the feet of Welsh and a terrific drive rocketed into the roof the net past the right hand of Keiren Westwood.
The second half saw chances at both ends. Welsh could have had a second in the 52nd minute, but fired across the face of the goal and David Poole brought the best out of Westwood.
At the other end Bridges and Karl Hawley were always lively, but it was midfielder Chris Lumsdon who had the best opportunity in the 86th minute.
He had a free header at the far post but nodded it straight into the arms of a grateful Mildenhall.