The men from Johannesburg were still in it at the break, when Ulster led by 12-10, but the hosts pulled away in the second half and scored 14 second-half points to secure the bonus point win.
Ulster had their first score after just three minutes when a short lineout throw to Andrew Warwick allowed him to find Timoney, who had peeled around.
Nathan Doak missed the conversion but instead of pushing on, several errors allowed the Emirates Lions back in the game and on 17 minutes they worked their way over the line through Stean Pienaar with EW Viljoen converting to put the visitors ahead 7-5.
After 34 minutes Ulster seized back the lead when a multi-phase attack resulted in Hume crossing the line with a man spare after the Irish province had simply worn the Lions’ defence out.
This time Doak converted and Ulster now led 12-7 but the half ended with the Emirates Lions closing the gap to 12-10 thanks to Viljoen’s long-range penalty.
After Will Addison was stretchered off early in the second half, following a collision with a Lions player, Ulster had try number three sewn up when a tap and go resulted in Rea being driven over the line with Doak converting again.
Ulster then should have had their bonus point when Tom O’Toole’s pass to Alan O’Connor sent the lock charging for the line only to be held up short with Doak trying to wriggle over when a pass ought to have been thrown.
The bonus point still looked inevitable and came when Timoney dotted down for the second time from a close-in drive with Doak converting.
The hosts enjoyed 58% possession and 59% territory, which had the Lions under pressure throughout the encounter.
Ulster 26 (12) - Tries: Nick Timoney (2), James Hume, Matty Rea. Conversions: Nathan Doak (3).
Emirates Lions 10 (10) - Try: Stean Pienaar. Conversion: EW Viljoen. Penalty goal: Viljoen.