Wolves, who have now won 12 of their past 13 home games and five games in a row, held out until six minutes before the interval when Pons leaped highest to grab a Thomas Lacans kick and touch down.
However, the hosts made the most of a rare attack in the final minute when Webster rounded off a lovely team move to skillfully score in the right corner to leave the game at 8-8.
Toulouse upped the urgency in the second half and another sustained spell of pressure resulted in Romeo Tropis crossing.
But just when it appeared as though the French side’s greater experience and nous could see them start to dominate, Warrington dug deep and showed their physicality.
Wood dived over from close range after taking a sharp pass from Lacans in his stride before Lindop pounced on another error at the back to gather the ball and cross.
And Josh Smith rounded off a wonderful night for the home side when a Hayes kick through bounced favourably and he was able to collect the ball cleanly and touch down.
Wolves will face Hull KR in the second eliminator on Saturday, 19 September, after Wakefield Trinity host Leigh Leopards on Friday, 18 September.
Warrington: Taylor-Wray; Webster, J Smith, Lindop, Thewlis; Irwin, Hayes; Yates, Powell, Harrison, Holroyd, E Smith; McKler.
Interchanges: Wood, McKinney, Hartill, Coop.
Toulouse: Polselli; Pons, Tropis, Jussaume, Ulberg; Lacans, Rouge; Belmas, Hands, Roumanos, O’Kane, Grossemy, Cator.
Interchanges: Marion, Chan, Butler, Garrigues.
Referee: Tom Grant.