The midfielders have both agreed three-year deals at Dens Park.
Williams, 28, moved to the Highlands in the summer of 2013 from Kendal Town and has made 35 appearances this term.
Former Kidderminster player Vincent, 26, also joined in 2013 and scored the winning goal when Caley Thistle won the Scottish Cup last season.
Vincent has been troubled by injuries this season and has been limited to 16 outings, the last of which came on 2 January.
"It's two boys we've been chasing for a long time," manager Paul Hartley told BBC Scotland.
"They've been part of a very successful Inverness team over the last couple of seasons.
"Danny is versatile, he can play in a number of positions. It's the same with Vincent, anywhere across the middle."
Meanwhile, Paul McGowan and Darren O'Dea have signed new contracts at Dens Park.
Midfielder McGowan, 28, has agreed a two-year extension, while former Celtic defender O'Dea, 29, has earned a three-year deal, having joined the club in January.
"We've been trying to build a squad over the last two years and this summer there will only be maybe three or four changes," added Hartley.

The firm said the system would allow its better drivers to receive cheaper premiums.
It follows similar efforts by smaller insurers. Larger rival Direct Line has told the BBC it is also piloting its own "black box" scheme.
Critics of the technology said that data should not be used as a reliable measure of a driver's ability.
The system involves the installation of a small black box into the driver's car which records how they drive.
The measures include monitoring speed, braking severity, cornering and the types of roads used during certain times of day.
This information is transmitted remotely to the insurers, and can also be accessed by users via a website which gives information on overall performance, warning them if they are likely to be moved to a higher premium.
"The reports are pretty detailed," AA spokesman Ian Crowder told the BBC ahead of Wednesday's formal announcement.
"The point is that these sorts of devices firmly put in the hands of the driver a responsibility for driving safely. It makes you think."
The information could be used to prove who was at fault in accidents, Mr Crowder added, but such detailed information would only be disclosed with a court order.
He added that the system could also detect sudden hard braking so assistance could be sent.
Extreme speeds would be greeted with "a stern email" to the driver.
He said the "pay-how-you-drive" system, which is aimed primarily at young drivers, could save customers up to Â£850 per year.
"All the anecdotal evidence suggests that people who have installed the system have about a 30% better claims experience - in other words, less crashes - than those who don't," he added.
The technology, known as "telematics" or "black box insurance" has for several months been utilised by smaller insurance firms such as Co-Op and Coverbox.
Welsh insurance firm Motaquote has just launched a partnership with leading sat-nav manufacturer TomTom to offer real-time feedback on driver performance.
Other major insurers are expected to launch policies soon. Direct Line told the BBC it had begun a pilot using its own technology - called Tracker - which it hoped to implement by the end of the year.
Elsewhere, car hire firm Avis said telematics is "one to watch", and said it might consider using the technology in its vehicles.
Malcolm Tarling, from the Association of British Insurers, said it is an approach which is likely to become commonplace.
"It's particularly important for young drivers who have high premiums," he said.
"You may say you don't want a 'spy in the car' as some call them, but others may say that if this is one way of making my premiums reflect my safety on the road, this will be of interest."
But Keith Peat, a spokesman for the Association for British Drivers, told the BBC he was worried that drivers who did not want to allow telematics in their vehicle would face higher costs.
"Providing the drivers give their consent it is OK," he said.
"But what we are totally against is people who don't give their consent being penalised."
Inverness Caledonian Thistle duo James Vincent and Danny Williams have signed pre-contracts to join Premiership rivals Dundee next season.
[NEXT_CONCEPT]
The AA is set to launch a new insurance policy which uses sat-nav technology to track driver performance.