Mourinho's initial spell at Stamford Bridge began with a couple of blistering openings to Premier League campaigns.
At the time, Sir Alex Ferguson's strategy was to start slowly and build momentum ahead of a flying finish.
It meant he was caught off guard as Chelsea flew out of the blocks and United were never able to make up for the lost ground.
Ahead of next Monday's encounter between the two clubs at Old Trafford, Giggs has recalled the experience, which should focus the mind given the trips to Liverpool and Manchester City.
"When Mourinho was at Chelsea, they obviously got off to good starts and we didn't manage to catch them," said Giggs.
"It has been well documented that we have a tough start but you like to get off to a good start.
"We got a good win at Swansea on Saturday and hopefully we can carry that on."
Giggs was speaking at the launch of a new sponsorship between United and India-based Apollo Tyres, who will become the club's official tyre partner in the UK.
There is a novel aspect to the deal in that Apollo, through their "Go The Distance" campaign, have vowed to create football pitches for community use made out of recycled tyres.
They intend for the first of those pitches to be constructed close to Old Trafford, with others intended for use in both the United Kingdom and India.
It represents the company's first venture into a global football sponsorship, having previously been involved in tennis, but not, interestingly, cricket.
"Cricket is a great sport but it is over-crowded," said Apollo's vice-chairman and managing director Neeraj Kanwar.
"First we went into tennis. We wanted to create Grand Slam winners in India. But we didn't see that being successful.
"It made sense to try a new sport. In India football is up and coming and there is interest in the Premier League.
"Manchester United have 46million fans in the country and we believe our aims and aspirations are the same as theirs."
United's more immediate aspiration is to build on their impressive weekend win at Swansea during a period that also includes trips to Liverpool and Manchester City.
Yet, whilst most pundits have tipped Chelsea to win the Premier League title at the present champions' expense, much to Rio Ferdinand's chagrin, it apparently is not the game he wants to win the most.
"You look forward to playing every time you step out at Old Trafford," he said. "It is a special place to play.
"We all appreciate playing there in front of our home fans and we try to put on a show.
"But the Liverpool and Manchester City games are usually hyped up a bit more than anyone else, so the feeling after winning one of those games is probably the biggest of the season."