Resuming on 55-1 in their follow-on and needing 100 runs to make Lancashire bat again, the hosts began badly.
Mark Stoneman (27) was removed in Stephen Parry's second over of the day.
But left-handers Sangakkara (136) and Borthwick (108*) joined forces to help Surrey reach safety on 323-3.
Sangakkara, dropped at slip when he was on 72, raced to a 154-ball century, while his old Durham team-mate Borthwick, who came south to Surrey over the winter, by comparison played second-fiddle.
But he remained unbeaten at the end, having completed his first century for Surrey, after Sangakkara's eventual removal, bowled by Simon Kerrigan just before hands were shaken.
Lancashire take 13 points from a second successive County Championship Division One draw from another game in which they had the upper hand, having also been unable to see out victory last week against Essex at Chelmsford.
Apart from their two match-saving centurions, they owed most to another excellent effort from fast bowler Mark Footitt, who has now claimed a haul of five wickets in an innings in five successive Championship matches for Surrey.
Having ended last season with 21 wickets in three matches, he has already taken 13 in two games this season.
Surrey centurion Scott Borthwick told BBC Radio London:
"This was something I've dreamt of for the past five months. It was a good feeling.
"We needed it as a team to bounce back from day three. Batting with Sanga and watching him go about his business in building an innings was unbelievable.
"It was the best seat in the house with him batting at the other end."
Lancashire coach Glen Chapple told BBC Radio Manchester:
"We had a very tough first hour in the game but, from that point on, have fought hard for four days and I'm really pleased with the performance.
"To make 470 on there after losing those five early wickets was a great effort thanks to that fantastic partnership between Jordan and Shiv Chanderpaul.
"We can take good confidence from here and the fact we've taken 12 bonus points from our first couple of games."
Kumar Sangakkara and Scott Borthwick shared a 256-run third-wicket partnership, batting for most of the final day together, to help Surrey escape with a draw against Lancashire.