McColgan triumphed on Wednesday evening in Birmingham – 36 years after her mother Liz claimed the same title at Edinburgh 1986.
Ms Sturgeon tweeted: “What an absolutely amazing night for @Team_Scotland - congratulations to all our medal winners so far in these #CommonwealthGames.
“Every medal is special, but what a tear in the eye to see @EilishMccolgan match her mum’s 1986 gold.”
Scottish Labour leader Anas Sarwar said: “Incredible! What a story!”
Channel 5 newsreader Dan Walker said his “whole family were screaming” for the last 600 metres.
Wrapped in a saltire, McColgan shared the emotional moment with her mother and her partner who were cheering her on from the stands in Birmingham.
Speaking to the BBC afterwards, she said: “It’s just been such an up and down year with Covid, initially, then another illness and a couple of niggles at the wrong time… but I honestly couldn’t have asked for any more tonight to have my family here.
“The crowd, in that last 200 metres, I can’t even explain it. It was vibrating through my whole body.
“I’ve never sprinted like that in my entire life. Without the crowd, I wouldn’t have finished like that. I just wanted it so bad.
“It’s just an absolute dream.”
Liz McColgan joined her daughter and said: “For me as a mother, not even as just a coach, to witness your daughter winning is just amazing and to win it in the same event I won it in.
“She’s just ran the race I always knew she was capable of winning. It was amazing to watch, it was very nerve-racking to watch. It’s been a long time coming and she put it together tonight and I’m just really, really pleased.”
Team Scotland won three other gold medals on Wednesday: Duncan Ross won his fifth medal for the men’s 200-metre medley, Sarah Adlington for the 78kg and higher women’s judo, and Rosemary Lenton and Pauline Wilson for the women’s para lawn bowls.
McColgan always destined for running
Eilish McColgan has overcome early career problems to flourish on the track and road.
The Dundee-born Hawkhill Harriers star was always destined for a future in athletics, having had the advantage of being the daughter of one of the greatest female athletes Great Britain has ever produced.
But McColgan, now 31, spoke this year of being perturbed by the sheer volume of promising young female athletes who quit sport in their teens.
In an attempt to keep young female athletes in the sport, McColgan and her partner, English middle-distance runner Michael Rimmer, set up a non-profit body, Giving Back to Track, to encourage continued participation.
"It's something I've been thinking about for a long time,” she said. “Every time I go home to Dundee I'm reminded of the great facilities we have now, and how much of a difference that makes for young athletes looking to progress, but with that comes increased cost or participation and competition, which can put people off.
“Once you start to factor-in travelling to competitions, accommodation and so on, I guess I've just become very aware that for some families it's virtually impossible to do all that.
“I'm in the fortunate position now where I can give something back to the sport and help people participate and compete. We're starting at home because that was really important to me. I hope it will make a difference to people and that we can build it up over the next few years.
“I've been there and done it - injuries, surgery, the ups and downs - and know how challenging it can all be. I think it's important to be honest in sharing my experiences. It's not all roses and unicorns, there's good and bad, but for me the enjoyment factor far outweighs the difficulties.”