The Lions had elected to bat first and had started solidly, Gulam Bodi (44) and Rassie van der Dussen (21) taking them to 80 for one after 19 overs. But the introduction of off-spinner Eden Links would see a collapse as the hosts slipped to 123 for five.
Conway and Dwaine Pretorius (32) revitalised the innings as they added 51 for the sixth wicket off 64 balls, before Links returned to trap Pretorius lbw in the first over of his second spell.
The left-handed Conway completed the recovery, however, batting through to the end of the 50th over as he faced just 73 balls and hit seven fours and two sixes.
Wicketkeeper/batsman Nicky van den Bergh (17) also proved a reliable partner as he helped Conway add 51 off 52 balls for the seventh wicket.
Crucially, Conway scored 20 runs in the final over, hammering left-arm seamer Vincent Moore for three fours and a six.
Links was outstanding, obtaining sharp turn and varying his pace well as he took four for 35 in 10 overs, while fellow spinner Roelof van der Merwe bowled the in-form Temba Bavuma, playing in front of an expectant crowd made up largely of children from half-a-dozen local schools, with a superb delivery that pitched outside leg and hit off, while conceding just 30 runs in his 10 overs.
The Titans were in awful early trouble as they slumped to 28 for three and opener Heino Kuhn had to retire hurt with a hand injury.
But Farhaan Behardien (60 off 76 balls) and David Wiese (94 off 102 balls) added 113 off 131 balls for the fourth wicket and it looked as if the crowd would go home disappointed.
But Pretorius, coming off a poor 2013/14 season, changed the complexion of the game by having Behardien caught at wide mid-on.
The Titans went into the last 10 overs on 197 for four, needing just 62 runs, but they suffered a mortal blow in the 41st over as Wiese was run out trying to take a second run to Eddie Leie at deep backward point.
Heinrich Klaasen, who had played second fiddle to Wiese during their stand of 58 off 56 balls, went on to score 43 not out off 47 balls on his franchise debut, but he was unable to find the boundary during the tight closing overs from Hardus Viljoen and Pretorius.
The Titans needed 22 to win from the last three overs, but the 48th over, bowled by Pretorius, went for just four runs and Viljoen conceded only five in the penultimate over.
Fast bowler Viljoen was the best of the Lions bowlers, taking one for 29 in his 10 overs and proving a handful throughout.