Begbie has been head of girls' and women's football at the Scottish Football Association since 1998, having previously occupied a similar position at sportscotland. She will become the Scottish Rugby Union's first head of women's rugby from August.
An accomplished footballer, Begbie made her Scotland debut when she was just 15 and played in the same national team as Rose Reilly. After scoring against Italy in the San Siro in 1974, Begbie was offered professional contracts in the USA and Italy, but turned them down.
Highly regarded within the SFA and Uefa, the 56-year-old will have to resign from her position as vice-chair of the Uefa Women's Committee to take up her new role. She was also a Uefa match delegate and was in Lisbon for last week's women's Champions League final when Wolfsburg defeated Tyreso.
Working in tandem with Scotland coaches Vera Pauw and Anna Signeul, Begbie helped to oversee the transformation of women's football from a marginalised sport to one which has risen in stature alongside Scotland's rise to 19th in the Fifa world rankings.
The under-17 and under-19 girls' sides both qualified for their most recent European Championship finals, while Scotland's leading club, Glasgow City, regularly reach the knockout stages of the Champions League.
"I am proud we have significantly increased the number of women and girls playing football, and with the help of an increasing number of clubs and volunteers there is now a great foundation on which to build the game in the future," said Begbie. "Nothing would give me more pleasure than to see Anna qualify for the finals of a major tournament with our women's team."
Stewart Regan, the SFA chief executive, has worked to eliminate much of the prejudice against women's football which once existed inside the organisation. "Sheila will be a huge loss. Her drive, expertise and connections across sport and politics have helped the women's game take huge leaps forward," he said.
The SRU will hope that she can reproduce this level of progress at Murrayfield. The Scotland women's team finished a dismal bottom of the 2014 Six Nations table, scoring just five points and conceding 261.
"What I'll bring from football is a blueprint of success," said Begbie. "I hope to transfer a lot of skills in terms of the strategic element of the work I've done at the Scottish FA and also the network of people I know."
Scott Johnson, the SRU director of rugby, believes her appointment will engender a bright future for the women's game. "This shows our intent to make women's rugby a priority," he added.