Lincolnshire Police launched an urgent appeal to find Daniel Boulton, 29, after the woman and child were found dead at a property in High Holme Road just before 8.30pm on Monday.
Boulton was detained by officers in the Hubbard's Hill area at around 12 noon on Tuesday.
He remains in police custody and is being questioned in connection with the murder of the woman and child - who were both found with multiple stab injuries. He is also being questioned on suspicion of injuring an off-duty police officer.
Another young child was found uninjured at the property.
Lincolnshire Police’s Head of Crime, Detective Chief Superintendent Andy Cox, said: "This has been a tragic, extraordinary, and worrying set of circumstances in equal measure. A woman and child have lost their lives in the most devastating way and I know that their loved ones will be feeling the magnitude of that today and our thoughts go out to them.
"I also want to thank our officers and other emergency services colleagues. The first responders last night were met with an unimaginable scene and reacted with professionalism to ensure we do what’s right by the victims, and that’s not a situation you can ever really prepare fully for.
“Our off-duty officer today showed how you’re never really off shift and how our staff will always do their very best to protect the public. A thank you doesn’t really cover it, and we wish him a speedy recovery.
“Finally, to all the officers and staff who have been involved in this challenging job overnight and into today, thank you for your ongoing commitment and professionalism.
“There is still a lot to do and now the work starts to understand exactly what has happened here and how.”