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, which 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.