Durham 223 (Trevaskis 88) and 364 for 3 dec (Dickson 186*, Lees 105) drew with Sussex 538 (Pujara 202, Rizwan 79, Alsop 66, Haines 54, Clark 50, Trevaskis 5-128)
They put on 313 for the first wicket with Dickson making 186 - his highest score for Durham - and Lees 105 before both fell in the same over from leg-spinner Mason Crane.
By then they had knocked off all but two runs of their first-innings deficit of 315 and when the players shook hands at 4.50pm Durham were 364 for 3 and leading by 49. Sussex take 15 points, Durham 10.
It was an outstanding effort by the Durham pair whose partnership was also the county's highest for any wicket against Sussex.
Lees' highest score in three Tests when he made his England debut against West Indies in March was a modest 31, but the left-hander faced 450 balls in that series and once again showed his powers of concentration here.
The pair did a lot of the hard work needed to secure the draw by getting through 51 overs on the third day unscathed and the only chance either offered before they were parted came midway through the morning session when Lees was on 67 and mis-timed a pull off Henry Crocombe only for Crane to spill a regulation catch on the fine leg boundary.
Ali Orr got his hand to the ball but couldn't cling on to a very difficult chance at short leg when Lees was on 97, but shortly afterwards he reached his 19th first-class century, made in three minutes shy of five hours, to add to the unbeaten 182 he scored against Glamorgan earlier in the season.
Sussex employed five bowlers in the pre-lunch session, but a slow pitch offering minimal lateral movement and only slow turn gave them little encouragement even when the new ball was taken immediately it became available.
Lees hit a six and ten fours and faced 262 balls and it was a surprise when he smashed a full toss from Crane straight to deep mid-wicket. If that wicket owed a bit to good fortune Crane was delighted when wicketkeeper Mohammad Rizwan held a catch at the second attempt after Dickson got a thin under-edge cutting later in the same over. Dickson's 186 included a six and 20 fours and was his second hundred of the season.
Keegan Petersen and Scott Borthwick took their side past the first target to make Sussex bat again before persistent drizzle forced the teams off shortly before 3pm. They returned an hour later and Durham lost a third wicket when Borthwick pushed forward to Delray Rawlins' left-arm spin and was athletically caught at slip by Rizwan, who had handed over wicket-keeping duties to Ali Orr by then and bowled a couple of overs of medium pace before the game ended.